The appearance of Rufus Wainwright in Edinburgh's much-loved and missed La Belle Angele in the Cowgate in October 1998 has passed into legend.

If everyone who claims to have seen him then was really there, it would have rivalled the Usher Hall (where he appears on Thursday) in size. The venue, destroyed by a fire that also took out the Gilded Balloon and The Bridge Jazz Bar, had its share of unforgettable nights in its relatively short existence (Jeff Buckley had graced its stage four years previously), but none who were actually in the audience will forget that first experience of Wainwright, then in his mid-twenties and one of the campest musicians many had ever seen.

He was on BBC Two's Later With Jools Holland that same month – on a show that also featured Marc Almond, Courtney Love's Hole and Red Snapper – but that appearance barely hinted at the live experience. Rufus told tales of his childhood with sister Martha and his youthful adulation of Franz Schubert. As well as much of his eponymous debut album, he sang Cole Porter's Miss Otis Regrets in a way that showed a familiarity with the interpretations of many jazz divas who had tackled that tragi-comic tale before him. The Herald's Rob Adams concluded his review with the simple "He's a one-off" and spoke for many with that assessment.

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With the benefit of hindsight, of course, what we saw then was the template for much of what was to come: the openness about his remarkable family and the tensions within it; the classical training and aspirations to create big pieces of art music; and the diva tendencies that culminated in a recreation of a concert by Judy Garland. With Rufus Wainwright – and it's a strain that runs through his entire clan, if not always quite as theatrically – what you see is what you get. And we don't often get quite that much.

Wainwright is on a huge tour with his self-consciously "pop" album, Out Of The Game, released earlier this year, produced by Mark Ronson and featuring the Dap-Kings, the musicians who backed Amy Winehouse on her most successful recordings. The Usher Hall date is followed by another at Glasgow's O2 Academy the following night – an indication of the fanbase he has built in Scotland since that first appearance in Edinburgh.

He remembers it well: "There was a very Gothic feel to the environment, like cathedral vaults – it was one of those. But I know Edinburgh and Glasgow because I'd been to both of those cities with my mother. She'd been singing with Emmylou Harris and taken part in the Transatlantic Sessions with Aly Bain. Is Dick Gaughan still playing?"

Wainwright is in Portsmouth, New Hampshire, and has a gig the next night in Montreal, where he studied classical piano at McGill University. But the reach of Celtic Connections is all embracing. I assure him that Gaughan, a man whose political songs the often-outspoken Wainwright admires, is indeed still on the circuit. And then I remember accompanying his father Loudon Wainwright III, with whom he has not always been on good terms, to a behind-closed-doors appearance at the Special Unit in HMP Barlinnie during a Mayfest in the late 1980s, and watching as Wainwright Snr listened as attentively to the songs the inmates had written as they had done to his. The world seems less large than it had earlier in the day.

Rufus Wainwright's mother, Kate McGarrigle, died nearly three years ago after a battle with cancer. When he plays Montreal, he tells me, her sister, "Aunt Ann", will be joining him on stage, and probably his cousin too. Sometimes his bassist is Brad Albetta, when he is not out on the road with his wife, Rufus's sister, Martha. His half-sister, Lucy Wainwright Roche, has been supporting on the tour, and in Scotland, support will be provided by Adam Cohen, son of Leonard and sister of Lorca, who is the mother of Rufus's daughter, Viva, who will be two in February. Oh, and Wainwright married his partner, arts administrator Jorn Weisbrodt, in August this year.

We know all of this because the last thing Wainwright is is shy. His life is so open it is a brazenly political act, although it is clearly equally a family trait. His publishing company is Put Tit On Music, which not only illustrates the sort of word awareness that eluded the Simon Cowell publicist behind Ms Boyle's #susanalbumparty, but is presumably a nod towards the song his father wrote and recorded for him when he was baby: Rufus Is A Tit Man.

Loudon has been more prescient in his songwriting, but that has not discouraged his son from penning songs for his new daughter. One of them, Montauk, named for the upstate New York home he shares with Weisbrodt, appears on the new record and imagines the day a grown Viva comes to visit her dad – and her other dad. The album's closer, Candles, recounts the mission of lapsed Catholic Wainwright to light a candle for his sick mother while on tour in Europe, and features Anna McGarrigle and Wainwrights Loudon, Martha and Lucy in a family chorus. The sharing is relentless, and yet it never seems too much. In Candles, incidentally, lay the source of that quote about "vaulted cathedral halls" – the song, and the record, plays out with a lament on the bagpipes, unlikely though that may seem on a Mark Ronson record.

After some fraught times in the family and youthful assertions of independence, Rufus Wainwright is now appreciative of the benefits of the family firm: "These are tough times for everyone. It is fantastic to be able to pool resources – it is not a publicity stunt." He jokes about having to find a new bass player when Albetta departed for Martha's tour: "I released him – I'm very diplomatic – because we do have separate careers."

The Out Of The Game tour still features star names Teddy Thompson – son of Richard and member of another dynasty – and Krystle Warren as part of an eight-piece band which is "more rock'n'roll" than some of his previous outings. Last time he came to Glasgow, for a tour based around the Songs For Lulu album, Wainwright stripped the music back to voice and piano for a show that was nevertheless well costumed and visually enhanced (by specially commissioned video work by Douglas Gordon).

Wainwright clearly compartmentalises his work. He is writing another opera to follow Prima Donna, which was commissioned and dropped by New York Met before being premiered in Manchester to a mixed reception. However, he is intensely proud of Songs For Lulu: "The piano and vocal arrangements were head and shoulders above most pop composition for voice and piano. I like to keep that quite separate, and that was the most amazing and intense tour for me, solo. It was time to chill out for this record, take it easy. I felt the need to lighten up and enjoy the ephemeral quality of the time I'm in. So lock up your daughters – and sons."

That meant turning to someone who might provide him with a hit album, and Ronson's way with Wainwright's tunes gave him a top 10 album in the UK, his biggest commercial success since 2007's Release The Stars, which got to No 2. For Out Of The Game, Wainwright dug out some songs that were as much as 15 years old, with the aim of making them generate some money to fund his next new project.

"There are key pieces that we had to excavate, songs like Bitter Tears that were overlooked or picked on by earlier producers, so I feel vindicated. You want all your children to do well. But no-one is buying albums any more and I am pleased that I can still make a success of touring the States from Little Rock, Arkansas, to Atlanta. The venues are not overflowing with adulatory young people – in Europe people are more willing to enlighten themselves – but I can sell 700/800 in small towns in the US. And I am an unusual artist – a gay opera fan playing piano – so it is a miracle that I am known at all. The bottom has fallen out of the business really, but I think I have diversified enough to get a job when I need one."

Perfect Man, another of the remade and remodelled songs on Out Of The Game, starts with the line "After another production of 'The Flying Dutchman', I landed in Berlin" – a slice of imagined biography it is difficult to imagine anyone else singing. For the past 14 years, that singularity has been the defining characteristic of the family man with the portfolio career.

Rufus Wainwright plays Edinburgh's Usher Hall on Thursday and Glasgow's O2 Academy on Friday. Out Of The Game is out now on Decca.