He woke in the backseat of a school bus, not sure where he was, holding hands with a girl he didn't know. (1.2)
At the very beginning of the book, Jason doesn't know who he is. This exactly parallels the experience of the reader, who doesn't know who Jason is at the beginning of the book either. Identity in books is something that grows over time as you build up knowledge of characters through backstory and description. Rather than saying that Jason has lost his memory, then, you could say that he is magically aware that the book has just begun.
It was like someone had wiped his memory, and she was stuck in the worst "do over" of all time. She wanted to scream. Jason stood right next to her: those sky blue eyes, close-cropped blond hair, that cute little scar on his upper lip. […] And he just stared at the horizon, not even noticing her. (3.6)
Jason's loss of memory is also Piper's loss. When it turns out that their relationship never actually happened, she loses part of her own identity. Our identities aren't made just by ourselves, after all, but also through our relationships with other people.
Chiron the centaur folded his front legs and bowed to her, and all the campers followed his example.
"Hail, Piper McLean," Chiron announced gravely, as if he were speaking at her funeral. "Daughter of Aphrodite, lady of the doves, goddess of love." (10.142-144)
Piper is not pleased to find out who she is. Being Aphrodite's daughter is not just a new beginning, but also an end of sorts for her former life, which might be why Chiron sounds like he's at her funeral—you've got to die first before you can be reborn (even if, in this case, Piper's being reborn with makeup).
"Who am I?" Jason asked the she-wolf. "At least tell me that."
Wolves don't have much of a sense of humor, but Jason could tell the question amused Lupa, as if Jason were a cub just trying out his claws, practicing to be the alpha male.
You are our saving grace, as always. The she-wolf curled her lip, as if she had just made a clever joke. (13.25-27)
When Jason finds out his identity, it turns out it's part of his name (his last name is Grace). Oops—duh.
"What are you?"
"The son of Hephaestus," Leo said. "And I warned you I'd destroy you with fire." (24.99-100)
This is Leo's superhero moment, where he zaps all the Cyclopes with fire and construction equipment. Son of Hephaestus is sort of his superhero identity—it's not as catchy as Superman, but it works.
He liked regular Piper better—someone he could hang out with. But the weird thing was, he couldn't quite get that other image out of his head. It hadn't been an illusion. That side of Piper was there too. She just did her best to hide it. (26.69)
You could argue that Piper all glammed up is a fake, masking the real Piper. Or you could argue that Piper dressed down is a fake, hiding the real glamorous Piper. Jason's take is closer to the truth though: Piper is both glamorous and dressed down, and therefor not fake either way. She's just revealing a different part of herself, the way the gods can be Greek or Roman but are still who they are.
He felt like a broken machine himself—like someone had removed one little part of him, and now he'd never be complete. He might move, he might talk, he might keep going and do his job. But he'd always be off balance, never calibrated exactly right. (30.45)
After Festus is destroyed, Leo feels like he's a broken machine himself, with bits left out. Again, identity is in part about other people: who you are is at least partly made up of who you love—and, in this case, who you made. Festus is arguably Leo's child as well as his friend, which makes him an even more integral part of Leo's understanding of himself.
"If Jason is a bridge, what's he connecting? Maybe two different places that normally don't get along—like the air palace and the ground. You had to be somewhere before this, right? And Hera said you were an exchange."
Thalia murmured something like a prayer. "I understand now why Artemis sent me here. Jason—she told me to hunt for Lycaon and I would find a clue about Percy. You are the clue." (36.68-71)
Jason is an exchange for Percy, which means in part that Jason is Percy. His identity—main hero guy—is the same as Percy's, after all. He's not just replacing Percy at Camp Half-Blood, but also in the novel itself. In earlier books in the series, Percy is the star—now he's gone, and Jason takes his place.
If the Mist could affect their memories, could Jason's whole personality be an illusion too? If their friend wasn't their friend, and they were heading into a cursed mansion—a dangerous place for demigods—what would happen if Jason's full memory came back in the middle of a battle?
"Nah," Leo decided. "After all we've been through? I can't see it. We're a team. Jason can handle it." (47.34-35)
Leo is wondering how much of who we are is based on our past. If Jason remembers a new past, would he be a new person? Leo then decides that Jason wouldn't be a new person because of all they have been through together. Jason's new past will, Leo feels, be sure to trump his old past—or at least be somewhat consistent with it. Leo feels confident—not because memories don't matter to identity, but because they do.
"I'm a child of Rome, consul to demigods, praetor of the First Legion." Jason didn't know quite what he was saying, but he rattled off the words like he'd said them many times before. He held out his arms, showing the tattoo of the eagle and SPQR, and to his surprise the giant seemed to recognize it. (50.68)
As always, Jason is learning who he is along with us. Plus he's scaring a giant, which is cool.