Tangerine is about Paul's search for his identity. (No, not his ID, his identity. Way more serious, although involving fewer lines at the DMV.) Sure, he wants to remember what happened to him in his past, but it's more than that—he needs to know what he can do. He finds that out in his soccer matches, but more importantly, when he fights for Luis at the awards ceremony, and when he stands up to Erik and Arthur afterwards—and when he remembers what Erik did to him. Once he remembers, he can move forward. Whole. And yes, with ID card in hand.
Paul could have snapped out of it without having to wait for another traumatic experience, but he didn't want to.
Mr. and Mrs. Fisher never do find their own true identities