The first time I saw Jane Birkin in the flesh – 15 years ago, at a recording in London of Jack Docherty's shortlived Channel 5 chat show – she arrived on stage looking artfully dishevelled and carrying a plastic bag.

Someone in the audience actually tittered. But even her reception and her dowdy quotidian prop couldn't dampen the sense of who she was: an iconic figure whose life story, if you knew its details, could serve as a counterpoint to the old saw that London was pop culture's centre of gravity in the 1960s and 1970s. Paris rocked too, you know, as anyone who heard the infamous 1969 single Je T'Aime - Moi Non Plus would know.

These days Birkin is more likely to be spotted carrying one of the famously capacious Hermes totes designed for and named after her than a bulging hand-out from Happy Shopper. But the rest of the baggage is unchanged: the word "iconic" still swings along behind her, heavier now as time lends weight to her legend.

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"I never quite know what it means," she sighs when I mention the i-word. "For me it's Elizabeth Taylor and one can hardly imagine Elizabeth Taylor shuffling down the street with her old bulldog like I do."

Birkin, now 66, has been muse, model, actress and singer in her day. Drawing all those things together is the person of Serge Gainsbourg, the restless French genius who used pop music as an instrument to teach, tease and offend, and who was her lover for more than a decade from the late 1960s when they met on a film set. He died of a heart attack in March 1991, aged 62.

Birkin and Gainsbourg had been separated for 10 years by that point though he had continued to write songs for her and she had continued to record them. After his death, then, she became the keeper of his flame, touring and performing his work either in its original form or in different musical styles. "I thought I must keep them going, I must keep him alive in some way," she says. "So I suppose that's what I did do. It's not painful any more. You can smile while you're singing whereas at the beginning I couldn't at all."

In 2002, Birkin released Arabesque, a live album collaboration with a group of Algerian and Roma musicians which marked the 10th anniversary of Gainsbourg's death. She will reprise the project later this year for a concert in Marseilles, currently European Capital of Culture. Before then, she returns to the songbook, this time in the company of a group of Japanese musicians she met when she visited the country in the wake of the Fukishima nuclear disaster. This month she performs the first of only two British dates with them at The Arches in Glasgow, part of a tour which also takes in performances in Rome, Rangoon and Ramallah.

Fans of Gainsbourg's songs will be pleased to learn that Birkin considers their current incarnation to be among the best versions she has ever heard. "Serge is brought sublime glory through these orchestrations," she purrs. "They're absolutely beautiful."

Another change is her own reaction to them. Now the songs feel as if they have no reference to her at all, a liberating experience. "They could be songs by [Charles] Aznavour. I'd forgotten what it was to be standing in the studio singing Amour Des Feintes [from her last album with Gainsbourg, from 1990] with Serge looking so ill behind the glass. I'd forgotten what it was to sing Fuir Le Bonheur De Peur Qu'Il Ne Se Sauve [recorded after they split in 1982] when I'd lost him."

Accompanying Gainsbourg's songs for the Glasgow date are Birkin's memories in the form of a 2011 documentary she made for French television channel Arte. Called Souvenirs Of Serge, it consists of home movie footage shot at various points throughout the 1970s, mostly by Birkin. Occasionally Gainsbourg takes the camera – cue frenetic zooms, crazy angles and Carry On-style close-ups of Birkin's bottom – and there are further segments shot by Birkin's Old Harrovian brother Andrew, now a film director and screenwriter.

The clips show the couple at play in France, London, Marrakesh and Venice, which they would visit almost every winter, booking into the same hotel and staying for a month. Here's Gainsbourg at Disneyland in California and crossing St Mark's Square with a cigarette in his hand; here's Birkin dancing round a wheat field wearing only a pair of bikini bottoms. So chic.

We also see Birkin's parents – David, a flamboyant former war hero, and Judy, nee Campbell, a celebrated actress once feted by Noel Coward – and her children: Kate, by her first marriage to Bond composer John Barry; Charlotte, Gainsbourg's child; and Lou, Birkin's daughter by director Jacques Doillon. Birkin, in her inimitable breathy style, does the commentary. It's bittersweet and utterly captivating.

"I thought it would be nice to show another side of Serge," she says by way of explanation. "Joann Sfar's film had just come out [Gainsbourg: A Heroic Life, from 2010] and I thought Serge was much funnier than that and much more charming. So I thought why not share a few Super 8 things that I had filmed of him on holiday with the children, where you could tell how funny he was. It's impossible to explain how funny someone is unless you can actually see them goggling their eyes the way he used to. He was such a surprising individual. Also I knew him at his most beautiful."

Birkin and Gainsbourg are still fused in the French popular consciousness and probably always will be. She tells me about getting into a taxi in Paris one night with a more recent partner, "a divine author I was with for a very long time", and finding the driver turning to face her. "We miss him so much, don't we?" the man said simply. Everyone knew who he meant.

There is, she thinks, "no identity" for her then. "It's always mixed up with Serge. So I suppose one has to take that on the chin and realise that, well, yes, that's what it is when you go on singing somebody else's songs."

In truth there is life for Jane Birkin away from Serge Gainsbourg. She continues to act, most recently alongside Penelope Cruz in the Bosnia-set Twice Born, and has long used her celebrity as a soapbox from which to "waffle on" (her words) about political causes close to her heart. The French listen, too.

In 2009, for instance, she published an open letter in Le Monde calling on French oil company Total to pull out of Burma in protest at the detention of Aung San Suu Kyi. It won a response in kind a few days later. In 1995, three days after finishing a production of Women Of Troy at the National Theatre in London, she visited Sarajevo, then still under siege. Most recently she was to be heard agitating for media coverage of the case of Malala Yousafzai, the Pakistani schoolgirl shot by the Taliban for blogging about women's rights.

And at the very least she can live vicariously through her children and grandchildren, following Charlotte to Cannes, for instance, where she won the Best Actress Award in 2009 for her role in Lars Von Trier's controversial Antichrist. The 41-year-old is currently filming Von Trier's follow-up, Nymphomaniac, which is expected to premiere at the French festival in May. Lou, 30, sings and acts too, is the current face of fashion house Givenchy and recently appeared in US television series Gossip Girl – playing herself. And Kate, "the hidden daughter" according to the headline on one French interview, is now 45 and a noted landscape photographer.

So after two decades as Serge Gainsbourg's representative on Earth, Jane Birkin could be forgiven for telling the world it's time to lay down the torch and quit reworking that rich musical canon. "I don't think I'll do it again," she says finally, in that famous cut-glass accent. "Maybe I'll just sit around for a while." Do you believe her? Moi non plus.

Jane Birkin performs The Songs Of Serge at The Arches on January 29 as part of the Glasgow Music and Film Festival. The concert will be preceded by a screening of Souvenirs Of Serge,