It’s not beautiful, Renzo Piano’s latest building. Sandwiched between a busy highway, the High Line (an old elevated train track that has been turned into a green walkway and public park) and the few remaining cold-storage facilities that gave Manhattan’s Meatpacking District its name, the new home for the Whitney Museum of American Art is not a landmark that instantly pleases the eye. All awkward angles and irregular shapes, it is given some harmony by pale ribs of steel cladding and planes of smooth concrete blocks, but in a line-up of great contemporary buildings – or even of Piano’s best buildings – it would definitely be the wallflower, judged from the outside. Which is fine, because in this case, it really is what’s inside that counts.

Talking to me on the phone from his office in Genoa the week before its opening, Piano described the nine-floor $422 million building as ‘a machine for art’, and it is from the inside that you really start to appreciate the huge, light-filled temple to creativity he has made.

"The building was conceived primarily to provide artists with inspirational spaces, to imagine what they can possibly do here," the museum’s director, Adam Weinberg, said on the opening day last month. "It offers us opportunities that we’ve never had before: larger, more flexible galleries in which to show art; outdoor terraces for performance and sculptures; theatres for lectures, performances, panels, installations. I want to salute Renzo Piano for designing this building from the inside out as much as the outside in, putting art and the artists first."

Piano and his team have positioned windows to provide stunning views across the Hudson river towards New Jersey from the upper floors, with equally panoramic views over Manhattan on the opposite side. There are also side vistas down towards the glittering new Freedom Tower and the Statue of Liberty, and outdoor spaces that help tie the museum to the surrounding area. Performances on the roomy third-floor terrace, for instance, will be easily seen and heard both from the High Line one storey below it, and from the street below that.

It is at street level that you best appreciate how beautifully the building has been designed to fit the site. It shrinks to half the size of the upper storeys, its walls all floor-to-ceiling glass. This lobby will be free to enter, allowing passers-by to use the restaurant or shop and view the art on display here. But what Piano has also done is create a public square between the museum and the steps leading up to the High Line, widening a previously narrow street to open up glorious new views of the water.

Piano famously has no fixed style, designing according to the site and the purpose of the building, but creating new public spaces has always been central to his work. That is why his biggest London building, the Shard, is so surprising. It dominates the capital’s skyline while remaining inaccessible to most Londoners: only the elite can afford to stay in its expensive hotel rooms and dine in its upmarket restaurants; it costs £30 even to enjoy the views from the top. When I say this, he insists I shouldn’t judge until the project – which includes the redevelopment of the busy London Bridge railway station and its surrounding area – is complete.

"We are still working on the station," he explains. "It will be another two or three years. We wanted the building not to touch ground, so that it doesn’t take the land selfishly, and the land is given back to the community in some way. I think this is part of the story."

Piano was born in the Italian port of Genoa to a builder father. His breakthrough as an architect came in 1971, when he won a competition to design a new arts hub in Paris with his then partner, Richard Rogers. The inside-out design of thePompidou Centre, with its elevators and utilities all visible on the exterior, caused a stir when it opened in 1977, establishing the duo as the new bad boys of architecture.

He laughs when I point out that he is now part of the establishment, entrusted with huge projects such as a new campus for Columbia University in New York, a film museum for Los Angeles, a national library for Greece and a palais de justice in the suburbs of Paris, as well as major projects across China and India. Rogers has become a lord, and Piano has been honoured in Italy with the similarly lofty title of senator, and although they no longer work together, the two men have remained friends.

"Not long ago, I had a conference with Richard in London," Piano says. "The two old bad boys, together again. It was really funny: him a lord, me a senator. But I still feel the same. I refuse to believe I am 77!"

Architects tend to start their best work at a point when many other professionals are coming to the end of their careers, and Piano is busier than ever, dividing his month between his offices in Genoa, New York and Paris – where his second wife and younger children live. He has four children, ranging in age from 15 to 50. He considers his buildings to be like family too, and says he likes to check in on them. He lives close to the Pompidou Centre, and still pops in regularly.

"I go there every month, for lunch or just to visit," he says. "It’s about affection. You have to see whether they are happy or not. I’m not romantic, but it is true that when you finish a building you hope for it exactly what you hope for with your children. You hope the building will be happy. Which means making people happy."

Retirement isn’t something he ever considers, he insists. "Why should I? I enjoy it. This is the only thing I can do, and I can’t stop. It’s such a complicated profession that it takes a long time to learn. So for the first 60 years, you are learning. Then you should have another 60 years to work. But that’s a bit difficult."

At the opening, Piano declared in a generous, witty speech that "it takes a thousand people to make a building". Leading that number is Elisabetta Trezzani, the partner in Renzo Piano Building Works who heads the New York office, just opposite the Whitney. She started work on the museum project in 2005, when its board was already battered by numerous unsuccessful attempts to extend their existing building further uptown. She was there through the selection of the site and the 2009 recession that almost derailed the project, and finally oversaw the building work.

I had assumed the biggest challenge her team faced was the massive fifth-floor gallery, offering the largest display space in New York without a single interrupting column. She says that is not so difficult with good engineers and enough money. For her, the biggest technical challenge was the steel cladding, made of panels 8mm thick and up to 22m tall, some of them curved and twisted to fit the building. Only one company, in Germany, was able to manufacture them. Then special containers had to be made to ship them over and a machine designed to hold them straight while they were fitted to the sides of the building, to give it a distinctive ribbed look. Watching from the window of her office while the panels were hoisted into place – twisting and buckling in the wind – was, she says with a laugh, one of the more heart-stopping moments.

It was Trezzani who took the call when the Italian fashion house MaxMara, which was sponsoring the opening party, rang with a strange request. Would Renzo Piano Building Works like to design a handbag with them, to celebrate the museum’s opening? At first Trezzani and Piano were hesitant.

"We’ve never done anything like that, so we didn’t know how to do it," Trezzani says. "But then we thought it was the kind of special project we would probably never do again. So we said, if we do it, we want it to be a collaboration because we need your expertise, and we can try to pick out the elements of the Whitney that we think could be an inspiration for the bag." She smiles. "Also, me being a woman probably helped a little bit, because Renzo only had a general idea, at the beginning, what a bag for a woman needed to be!"

They eventually decided to echo the Whitney’s steel ribbing on the sides of the handbag, only in leather. "We wanted to do something very detailed and well done, and MaxMara were great to work with. It was fun for me," Trezzani adds.

MaxMara’s relationship with the Whitney goes back way before the new building was conceived. The company’s founder, Achille Maramotti, was a keen collector of art, a passion shared by his son Luigi, who is now the company’s chairman. Luigi lived in New York during the 1970s and 1980s, and over the years ties had been strengthened as the Maramotti family lent pieces from their collection for Whitney shows, or helped with fundraising  events. So when the museum called to ask them to sponsor the opening-night party, they agreed immediately, especially as the project also involved Italy’s greatest architect. "It was the perfect storm for us," Maramotti says. "It made sense."

Fashion, he says, is always exploring ideas about society, about modernity, pushing at the borders of what is acceptable. "But artists are doing that first, and probably better than fashion designers. We ultimately have to make something that somebody will buy and wear, but for an artist it’s different. It’s a necessity to express himself, and he would do it anyway, even if there were no buyers."

Like many big names in architecture, Piano has crossed paths with the fashion business before, designing a flagship store for Hermès in Tokyo in the 1980s that still looks glitteringly impressive today. For him, the main difference between his industry and the fashion world is speed.

"What is great about fashion is that it reflects the moment. It’s like taking a picture of a moment," he says. "Fashion relies on the short-term. Architecture is long-term. It’s about cities, longevity. That’s why architects need time to be accepted, to be loved. It’s not instant. Architecture often becomes loved when it is adopted by the city – when you go to the church, to the theatre, the university, the concert hall and it becomes part of your daily life. The great thing about fashion is that it is fun, an expression of joy, instant and ephemeral."

MaxMara, however, doesn’t really focus on the ephemeral. Its hallmark is timeless classics, pieces that will last and remain stylish, that can be passed from mother to daughter. The label’s creative director is Ian Griffiths, a genial Briton who has worked with the family for 27 years. He talks about the Whitney bag as a future classic, something he hopes the brand will still be making in 20 years’ time.

It is easy to believe he is right. A limited edition of 250 in the same pale grey as the museum’s steel exterior sold out on the first day, but it is a striking, elegant bag in any colour – boxy in shape and roomy enough for all your daily needs without being weighted down by heavy hardware that makes it impossible to carry that long. The Whitney’s ribbing is represented as pintucks in the leather, which posed a challenge for the label’s leather workshop.

"The pintucks were extremely difficult," Griffiths says. "And this is where working with an Italian company is so great. Our craftsmen in Tuscany use techniques that have been perfected over generations, but they are also very open to new ideas."

Just before the museum opened its first show in the new building to the public, I was given a private tour with Griffiths and a handful of other journalists. The result of three years of discussion among the curatorial team,America Is Hard To See is an unflinching and intellectually rigorous survey of 115 years of American art. It is also full of surprises, juxtaposing familiar pieces by renowned artists with less well-known works.

Many of the choices are brave – a large, bold piece by Donald Moffett attacking President Reagan’s failure to recognise the Aids epidemic, for instance, or a collection of harrowing works depicting lynchings. "I don’t think there can be many institutions  that would be prepared to be so direct about an issue like that," Griffiths says. "It explores facets of the American psyche as you would a person, and in any personality there are good and bad aspects."

His pre-fall collection for MaxMara is all Whitney-inspired, coming out of conversations with the head curator, Donna De Salvo. "I didn’t just want to start from a piece of art that was in the collection. What are you going to get out of that – scarves with a print inspired by Jasper Johns? It seems a bit shallow and random."

Instead, he read the American artist and intellectual Robert Smithson’s musings on memory, which led him to research the rich artistic and cultural history of the area around the museum, including his own memories of visiting the Roxy, a club that in the late 1970s and early 1980s was a playground for artists such as Andy Warhol, Keith Haring and Jean-Michel Basquiat, as well as the singers Debbie Harry and Madonna.

He also read Smithson’s thoughts on urban entropy, the idea that, inevitably, an urban space will degenerate back into a natural state. He connected this to the growing trend for rewilding and the creation of the High Line. Then he saw a 2008 installation by Fritz Haeg in the Whitney’s archive representing the return of animals that once would have lived in Manhattan such as the bald eagle, the barn owl and the eastern mud turtle – which, in turn, led to bold animal prints in his collection.

Given all of this, it is probably no surprise that Griffiths is also a professor, teaching fashion at Kingston University in London. At the age of 18, he thought he wanted to be an architect and did a year of study before he realised it wasn’t for him and switched to a fashion course. Still, he stayed in touch with friends he made in that year, most of whom qualified as architects. "I met up with some of them last weekend," he says. "I said, OK, you completed the course and I didn’t. But which of us is working with Renzo Piano?"

The new Whitney Museum of American Art opened to the public on May 1.America Is Hard To See runs until September 27. The Whitney bag, from £740, is available in the museum shop and MaxMara stores