Sitting with the literate and witty Regina Spektor in a Russian restaurant in midtown Manhattan, drinking tea with sour cherries, the conversation turns to the nature of her art. "It's mysterious to me," she says. The question was whether her songs are self-confessional, and if not, to what extent they involve some kind of emotional processing. "I'm more like a science fiction or fiction writer than a diarist or an essayist, but all those people are just as emotionally involved. I don't believe that someone who writes diaries or memoirs is more emotionally involved [in their work] than someone who writes fiction. I wouldn't believe that Kurt Vonnegut is not as personally involved as Anaïs Nin."

It's easy to see why fans of the 32-year-old singer-songwriter – whether the (female) colleague who told me "I've sobbed with heartbreak watching her" or the (male) colleague who admitted to holding a crush or perhaps even Barack Obama, who's seen her play live twice – might still want to parse her lyrics for a sense of who she really is. Five critically hailed albums into her career, with a sixth, What We Saw from the Cheap Seats, due later this month, she has developed a loyal fanbase, one quite as intense as she can also be in person.

The Spektor oeuvre is thick with different styles, ideas and allusions – even a chunk of a Pasternak poem sung in her native Russian on Après Moi on the 2006 album Begin to Hope – and the new record, which, she says, "feels like my best work to date", again essays several moods. There are quirky songs, like the urgent lead track All the Rowboats, political barbs, ditsy love songs and even the sort of potentially mawkish ballad – complete with sweeping strings – covered by contestants on TV talent shows. I tell her it's impressive that she's resisted the temptation to write more songs of that ilk. She replies: "There are lots of different places I like to go to explore. I've written a straight-up jazz standard type song, and I've written a country song, if produced in a colour-by-numbers sort of way. I'm not really interested in having a specific genre, though. I don't believe people have to commit to anything artistically." Sometimes, though, there's something about a song that can't be denied: if it wants to sound like a power ballad, it will do. "It's like you have a kid who's great at sports but you want them to be artsy and read books all day, like you did. You don't want to be going to their football games. But they're naturally an athlete, they're naturally a jock. They want to be in that culture. You can't deny somebody, or a song, its natural properties. But then at the same time you can push back and not put on the exact colour-by-numbers string section."

Spektor was born in Moscow in 1980 to a Russian Jewish family and seemed preordained to be a classical pianist: her father was a photographer and amateur violinist and her mother a professor at a Soviet college of music. The family left the country during perestroika, in 1989, when Spektor was nine and a half, and were admitted to the US as refugees with the assistance of the Hebrew Immigrant Aid Society. "There are so many things I wouldn't understand in the way that I do, or have connection with, had I not grown up there for half my childhood," she says earnestly. But she also notes: "I didn't have to deal with the reality that my parents had to deal with".

Settling in the Bronx, she continued her piano studies at the Manhattan School of Music and graduated early from a four-year course in studio composition at the Conservatory of Music at Purchase College. There was also a colourful stint working at a butterfly farm in Wisconsin, plus a further spell of study in Tottenham in north London. ("I loved it there," she says, even if "every time I'd leave Wood Green tube I'd see a sign that was like, 'Last night there was a stabbing on White Hart Lane.'")

Somewhere along the way she'd heard Joni Mitchell, and started writing her own songs, becoming a part of the blossoming "anti-folk" scene back in New York and touring with the Strokes. "I'm not built for classical music," she says. "You have to be built for classical music in the same way that you have to be built to be an Olympian."

Spektor's third album, in 2004, may have been titled Soviet Kitsch, and the new record recorded principally in LA, but she agrees that she is a quintessential New Yorker. "I don't feel in weird denial of my heritage. I don't feel like a born and raised New Yorker, but… I'm so influenced by this place. I feel so me here."

She's even been adopted by the president, having performed in front of Barack Obama at a reception in honour of Jewish-American heritage at the White House, and at a subsequent fundraising gala, for which – according to her manager – her presence was specifically requested. "I was totally excited," she reports. "He said: 'It's nice to see you outside the White House, Miss Spektor.'" She is a fan: "I obviously, like everyone else, wish that things could get done faster and I don't agree with all the policies, but I think on a human level... politicians like him are rare. We're lucky to have him as president. I can't begin to imagine what happens if he doesn't get elected again."

Nonetheless, I don't think she's necessarily a doctrinarian Democrat. "I'm a very liberal-minded person but I'm not a blanket anything," she says, adding of the other side of the party divide that "at its core they are really cool… before it got hijacked by all the fear-mongers and Bible stompers and all the scary people who made the Republican platform a platform of hate. There's an interesting fiscal [idea there] and other things that are good points. If there was just...

"I would be so excited if there was a real candidate who had real points," she continues. "I would be excited for that person – to give the Democrats a difficult run – just because they would be a worthy opponent."

Her upbringing appears relevant in this context. I put it to her that the worst thing that happened to the west was the collapse of the Soviet Union because it represented some kind of corrective to capitalism's excesses, however illusory that promise. "I think people forget very easily the kind of crazy human sacrifice that kind of system [involved]," she replies slightly stiffly. "Everybody was just trapped. There wasn't any place for any kind of growth. It's easy to romanticise it on a social level. There were positives that came out of it. Someone could grow up in absolute poverty and be very close to their family and they'd be very good people, but you can't romanticise it. It's not fair to romanticise it."

We meet just after Putin has reclaimed the presidency in Russia. "There wasn't any real person who could have gotten elected there either. It's sad. Our world is so full of brilliant people but they just don't get to be there for the most part," she says. "If they squeeze through, it's a miracle."

Spektor is a newlywed, but of her recent marriage to Moldy Peaches guitarist Jack Dishel she won't say anything; and nor, understandably, is she keen to discuss the drowning of her cellist Daniel Cho in Lake Geneva the day before she played the Montreux jazz festival in 2010, or the recent death of another close friend. But she does say, turning hushed and sounding understandably uncomfortable: "I'm definitely in the club of people who have experienced great tragedy in their life. Nothing bad had ever really happened to me but now I'm in this club – and it's a really big club. I don't think I was prepared for the level of pain I've been experiencing in the last few years. As you go through life you try to take all the things that come your way and process them with as much strength and kindness as you can muster. Obviously it transforms you as a person, but I don't think it's necessarily a final definition."

It's that question of definition that leads to the discussion of her work, and the impossibility of pinning too much biographical fact on the shifting surface of her frequently dazzling songs. This is someone who says her favourite novel, "a constant source of inspiration", is Pushkin's Eugene Onegin – but at the moment she's lapping up Bossypants, the essentially autobiographical book of comedian Tina Fey. It's a very Regina Spektor combination.

Concluding, she adds, not pausing for breath: "I still feel that art comes from a bigger place than just your own experiences and your own daily struggles or thoughts. A lot of it comes from a place of feeling rather than conscious thought."