Who is he? Well, he's two people: he's Peter Pan and he's the Doctor.
I love it. Peter Pan and Doctor Who have so much in common.
Peter stays forever young through sheer mind power: he thinks to himself that he is young and that is enough to keep him that way (movies tend to say that Neverland keeps you from aging, but in the original story other characters there do age and it's only Peter who doesn't). The Doctor regenerates, which other Time Lords did, as well, but now he's the only one left and so it becomes a singular, strange, and rare thing for him to do. And both Peter and the Doctor are very thrilled at what they're able to do.
The Doctor's arrogance is usually funny or endearing or at least something you're willing to put up with, but Peter Pan is probably even worse. Though he has managed to become a beloved character because he reminds us all of our childhood, he can be quite rude and irresponsible and so self-centered that he either annoys or is a danger to the characters around him--which is not always endearing. Then again, the Doctor often puts the people around him in danger, too. But they don't necessarily mind, do they?
That's what both characters offer: adventure beyond the scope of sight. Peter takes Wendy and her brothers to see Neverland and she is in awe of this land up in the stars. The Doctor invites Rose and all the rest to be his companions on his endless journeys through time and space, showing them worlds they would never otherwise have visited and time periods they never thought they could enter; most of them love it so much that they get almost addicted to it and have a hard time leaving.
But no one stays forever with Peter Pan and no one stays forever with the Doctor. Peter's friends all grow up: there's that (semi-disturbing, if you think about it) line in the book that explains that he finds a way to let the boys who are getting older die, or characters like the Darlings leave Neverland and go on to live adult lives in the regular world. It's the same for friends of the Doctor: they either die in one of the Doctor's dangerous adventures or go on to live essentially normal lives (some choose to leave and some are forced to by outside circumstances). It's the choice, tragic at times, that we all must face: do we try and make adventure last forever until it tears us apart, or do we put an end to it and live the life everyone else leads? Peter looks in on Wendy only to find she's an adult with a daughter of her own, and the Doctor meets Sarah Jane again only to find that she's over the hill--and that gap is so deep, despite the fun adventures these characters had together when both sides were young.
Peter wants to remain young because he keeps making himself so, but he does feel pain at moments like this: there is tragedy that he, unlike Wendy, will never grow up. But he accepts that cost. The Doctor, on the other hand, has no choice: all he can do is continue to live while everyone he has known withers away. Both characters are eternal--all characters are eternal because they live forever within the (metaphorical) pages of fiction, yet these two characters are doubly eternal because they are literally living forever within those literal pages, as well as the metaphorical ones (not that Doctor Who is a book instead of a TV show, and therefore the main bulk of the story isn't on literal pages, but I'm taking "literal pages" to mean the content of the story as it is presented to the audience and "metaphorical pages" to mean what the audience can always go back to revisit, whether through rereading, rewatching, or going back over in their heads).
Oh, yes, and when I said that both characters can fly, I was naturally referring to the TARDIS. Through it, the Doctor can fly--through both time and space. Peter Pan just literally flies through the sky. Two characters, forever flying away from the stable life they will never have--forever flying away toward adventure.