I would suggest that one way the Eliezer loses his sense of self in the narrative is through the idea that survival becomes the driving force of his consciousness. The brutal conditions of life in the camps help to remove Eliezer from aspects the previously defined his sense of self. The movement from Sighet helps to dislodge him from his physical community. The breaking up of his family helps to remove his emotional sense of self. The violation of his faith loses his spiritual compass. In the end, the Nazis end up reducing Eliezer to a being where survival is the only driving force of his state of being in the world. At the end when Eliezer cannot recognize the face in the mirror staring back at him, it is symbolic as well as a physical displacement that has transpired.
In the book Night Elie had always had a very spiritual side that he thought would help him through life. He and his family are taken off, separated, and he and his father are forced to endure the hardships of the concentration camps.
Elie slowly loses his self identity. Initially, he has lost the boy he had been. He then begins to lose his spirituality when he witnesses the young boy being hung. Later as he finds himself repulsed that his father is drawing attention to himself by his inability to keep up with work and becomes sick, Elie loses who he was as a son.