Your identity is what makes you "you." If you are having "an identity crisis," then I guess you can't figure out who you are. Good luck with that.
The noun identity can also refer to a name or persona. Criminals tend to use false identities so they won't get caught. If you falsely identify someone, it is a case of "mistaken identity." The word identity doesn't have to be used for a single person either. People refer to "corporate identity" when talking about what makes a company unique.