How we cite our quotes: (Chapter.Paragraph)
Quote #7
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.
Quote #8
"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.
Quote #9
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.