These factors can be divided into a few main categories:
His family is what formed his initial self-identity, through the expectations they had of him and how they interacted with him. His mother and father ran a store, and his two older sisters helped with that. Elie, however, is told that his place is in the house of study, to become learned and wise. This shapes his identity to one of curiosty regarding learning, and leads him to seek instruction in matters of religion and spiritual growth.
The next factor that influences Elie is the factor of his religion. Elie is a Jew, and as such is a follower of many traditions and is also certain to be judged by others; especially Nazis. His religion not only affects his self-identity, how he describes himself, but also his public identity, how he is judged by others. Elie describes himself as a devout Jew, dedicated to learning about God, and seeking to become One with Him through spiritual enlightenment. This influences the decisions he makes; such as accepting Moishe the Beadle as his tutor in the intricacies of Kabalah. Moishe in turn also influences and molds Elie's identity through the knowledge and understanding that he leads Elie to. Out of the three factors, Elie bases most of his life and decisions on the connection to his religion that he has.
The third factor that influences Elie's identity is that of the events that occur around him. First, Moishe is deported because he is a foriegn Jew. Then, he returns and warns everyone about the horrors that are being committed against Jews. However, Elie did not believe him, and sought to get him to just move on with his life and stop trying to warn others. This is Elie's first encounter with how crushing encounters with Nazis are, as Moishe tries to tell him that he wanted to return to try to warn the rest of the Jews so that they may "ready [them]selves while there is still time." Elsie doesn't understand what he means right now, but once he reaches Auschitz, he will understand.
These are the factors that are revealed to us as shaping Elie's identity in the first two chapters of Night. Doubtlessly, his identity shall continue to evolve as he goes through experiences in the upcoming chapters; much as our identities are always evolving.