How we cite our quotes: (Chapter.Paragraph)
Quote #1
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.
Quote #2
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.
Quote #3
"This isn't me," Piper protested. "I—don't understand."
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).