MCGONAGALL: This boy will be famous. There won't be a child in our world who doesn't know his name.
DUMBLEDORE: Exactly. He's better off growing up away from all that. Until he is ready.
Dumbledore's making a specific choice here that will help determine Harry's identity: a life of celebrity and fame? Or anonymous drudgery with the Dursleys? The seemingly crueler choice may actually make Harry a better person.
HAGRID: Blimey Harry, didn't you ever wonder where your Mum and Dad learned it all?
HARRY: Learned what?
HAGRID: You're a wizard, Harry.
Harry isn't aware of who he is, though everyone else seems to be. The story involves what he does that with knowledge far more than simply learning it…which suggests that a big part of his identity is his to choose.
PETUNIA: Of course we knew. How could you not be? My perfect sister being who she was. Oh, my mother and father were so proud the day she got her letter. We have a witch in the family. Isn't it wonderful? I was the only one to see her for what she was. A freak! And then she met that Potter, and then she had you, and I knew you'd be just the same, just as strange, just as...abnormal.
Aunt Petunia certainly has her own ideas about who Harry is. Lucky thing they don't stick.
HAGRID: See Harry, you're famous.
HARRY: But why am I famous, Hagrid? All those people back there, how is it they know who I am?
HAGRID: I'm not sure I'm exactly the right person to tell you that, Harry.
Again, the movie emphasizes Harry's Journey here as one of self-discovery—learning who he is—more than just stopping the bad guy.
HAGRID: But one thing's absolutely certain. Something about you stumped him that night. That's why you're famous. That's why everybody knows your name. You're the boy who lived.
On the surface, this is a little frightening. It suggests that Harry survived Voldemort's attack solely because of some fate or fluke, and therefore that Harry's identity is determined solely by outside forces. The more he learns, however, the more Harry starts to understand that he's still responsible for who he becomes…and that "the boy who lived" is just a label without a lot of meaning.
SORTING HAT: Hmm...difficult, very difficult. Plenty of courage I see, not a bad mind, either. There's talent, oh yes, and a thirst to prove yourself. But where to put you?
HARRY: Not Slytherin. Not Slytherin.
SORTING HAT: Not Slytherin, eh? Are you sure? You could be great, you know. It's all here in your head. And Slytherin will help you on your way to greatness! There's no doubt about that! No? Well, if you're sure...better be...Gryffindor.
Like Petunia, the Sorting Hat has some very specific ideas for who Harry is. Unlike Petunia, it actually listens to him, and assigns him a House based on his wishes, not its own.
MCGONAGALL: Potter, this is Oliver Wood. Wood, I have found you a Seeker!
Harry loves Quidditch. More importantly, he's very good at it: just another side of himself that he finally has a chance to discover at Hogwarts after eleven years of virtual slavery.
RON: I'm telling you, it's spooky. She knows more about you than you do.
HARRY: Who doesn't?
This is the downside to fame: everyone knows far more about Harry than he's comfortable with. It's doubly irritating since he's still trying to figure it all out… which isn't helpful when so many people know so much about him.
HERMIONE: You'll be okay, Harry. You're a great wizard, you really are.
HARRY: Not as good as you.
HERMIONE: Me? Books and cleverness? There are more important things. Friendship, and bravery.
A little humility from Hermione gives her know-it-all qualities some humanity. More importantly, she confirms who Harry is…and more importantly, confirms that he's becoming what he most wants to be.
VOLDEMORT: Yes. You see what I have become? See what I must do to survive? Live off another. A mere parasite.
Like a lot of great villains, Voldemort's vain, which helps define his identity even at this early stage. It also explains at least some of his anger at Harry: he's butt-hurt for being reduced to a face on a bald guy's skull.