What kind of characterization & identity does Gatsby have?

A self-manufactured one! He comes from a humble background, as we learn when his father shows up after his death, but he has created a pseudo-preppy persona for himself after making his fortune as a bootlegger. (E. g., he calls his male friends "old man.") The shirts incident reveals that he is really nouveau riche, but beneath that fact, he really does have some innate cl. Ironically, he appreciates Daisy more than does her supposedly blue-blooded husband, who is having an affair with his mechanic's wife, and ultimately he is killed for something Daisy actually did. You'll have to check the last chapters to be sure whether he knowingly lets Myrtle's husband think he was driving the car that hit her or whether he's killed so quickly that he doesn't have time to protest his innocence if he wanted to.