This is an interesting question because I would claim that McCandless never lost his identity. His identity changed as he embraced his wandering, individualistic lifestyle and self. His identity was just as vibrant and full as it ever was; however, it becomes somewhat different from what his friends and family members had come to know. I say somewhat different because McCandless always did his own thing, and his third grade teacher even picked up on this fact.
His teacher pulled us aside and told us that "Chris marches to a different drummer."
While McCandless was very comfortable doing his own thing, he was still subject to societal pressures. It's why he was a fairly good student and why he went to college; however, all that started to change when McCandless found out about his dad's affair. This is when his identity started to shift, and he really started to embrace a solo, loner persona. He did things like give away the rest of his college money and take extremely long road trips. McCandless was transforming himself into the wanderer that he probably always wanted to be, and he changed his name to help cement this fact in his mind.
Driving west out of Atlanta, he intended to invent an utterly new life for himself, one in which he would be free to wallow in unfiltered experience. To symbolize the complete severance from his previous life, he even adopted a new name . . . he was now Alexander Supertramp, master of his own destiny.
In McCandless's mind, he hadn't lost any of his identity. He had gained a completely new identity that was much fuller and more exciting than just being "Chris." People other than McCandless might say that he lost his identity and became a wandering nobody, but McCandless would not agree.