In the last years of his life Alberto Giacometti got friendly with a young woman called Yvonne Poiraudeau – a wild character who hung out with prostitutes and gangsters in the Paris of existentialism and New Wave cinema. Giacometti calls her Caroline in his portraits of her.

Paintings of Caroline hang together in the formidable final room of this exhibition, her features worked over in tough spiderwebs of intent drawing and suspended in brown and grey washes. This woman – who Giacometti once tried to get released from prison on a theft charge – takes on the dignity of Queen Nefertiti. She is lost in time, ancient and monarchical. These paintings seem to have been found, not made. Caroline takes her place among Egyptian statues and bog-preserved bodies as a sublime image of the enduring and irreplaceable mystery of what a human being is.

In postwar Paris, in a Europe that was still a raw graveyard, its earth scorched to ash by genocide, Giacometti was the only artist who rose to the occasion. He found something to say that was still meaningful after Auschwitz. He dug down to the simplest truth. In Giacometti’s greatest art, the human essence – primitive, unadorned, and somehow still vital after the most violent years in human history – becomes a totemic image of some final desperate hope: a slender chance.

One of Giacometti’s tall thin women, cast in bronze from a roughly modelled strand of clay, stands as static and archaic as the ancient Etruscan tomb statues that inspired her. Such figures by Giacometti stand in every museum of modern art. But only one of them is here. This is not a display of abstract mystery. It is an exhibition about people, emotions, loyalty and love: a revelation of the human passion out of which Giacometti created his enigmatic vision.

All too often the story of art – especially modern art – is told in a chilly academic way as if artists were barely human beings at all, as if their art was a lofty philosophical game remote from real life. In such tellings, what you need to know about Giacometti is that he started as a surrealist then rejected Freud and became a figurative artist. It’s true that Giacometti created some great surrealist sculptures, including his terrifying 1932 work Woman with Her Throat Cut.

But you won’t find that here. This show tells another story, centred not on the Paris avant garde (of which Giacometti was a star both before and after the second world war) but his intense relationships with his mother and father and siblings. It starts in Borgonovo, Switzerland where Giacometti was born in 1901. His first sculpture, created aged 13, is a bronze head of his brother Diego. Four decades later, he was still making sculptures of his brother. The amazing Bust of Diego (1955) puts a thin head atop a thin body, at contradictory angles: either you see a full profile over flat shoulders, or a pin-like head on a big bronze bottle of a chest. It is hilarious, like a Daumier cartoon of a pear-headed king, and it is profound. No one can be summed up in a work of art – Giacometti has been looking at his brother all his life but still can’t pin him down – people are indescribable.

Giacometti looked at his mother Maloja for much of his life, too. In his great 1947 painting Portrait of the Artist’s Mother she is a fading, almost spectral presence in a room of russet shadows, almost like a red-walled chamber in Pompeii. She is slipping away from the artist, into ancient time.

It is fascinating to see Giacometti’s early works – his first sculptures and paintings are influenced by everything from Donatello sculptures he saw on a trip to Florence and the sub-Cézanne style he got from his artist father – and then to meet him in Paris at the height of his greatness.

He was the same man, this survey reveals. Far from an avant garde sophisticate whose art reflects Parisian ideas, Giacometti was grounded in family. He always went back to Switzerland and he always portrayed his family alongside friends like the Communist poet Louis Aragon and the outlaw writer Jean Genet.

An obsession with the simplest act of recording what people close to him look like runs right through the show. It doesn’t matter if Giacometti is making a bust of his bearded father or drawing the tough sensual face of Genet. He does not use people as material – he tries to do justice to them. He is a truly faithful artist.

Out of that faith in people, Giacometti created an art of heroic ordinariness. The grandeur and comedy of his art creep up on you as you look at deliberately ludicrous bronze busts of his model Annetta that are, somehow, utterly beautiful and moving. Its realism is completely disarming in a pencil drawing of the collector David Sainsbury – you just know this is an Englishman, wearing George Smiley’s glasses.

Giacometti gets people just right, as they are, without pretence. Out of that honesty he created the most truly universal art of the past 75 years. This meeting with Giacometti the man makes his achievement all the more moving.

Here are the bronze survivors of a broken age.

• Giacometti: Pure Presence is at the National Portrait Gallery, London, from 15 October to 10 January 2016