'When I say to myself, "Bernard", who comes?
. . . They do not understand that I have to effect different transitions; have to cover the entrances
and exits of several different men who alternately act their parts as Bernard.'
This passage from Virginia Woolf's The Waves, encapsulates a notion which I feel is extremely relevant to the idea expressed in the prompt. To explain the quotation, the character of Bernard, as presented here by Woolf, epitomises the premise that when we are with other people we are merely acting out who they think we are, rather than who we really are. But if that is the case then how do we, ourselves, know who we really are. For this reason, Bernard in this passage experiences a crisis of identity, because he spends so much time playing parts to entertain other people, that he no longer knows what it means to simply be himself.
You might also want to think about how one's existence may become predominated by the existence of a complementary other, such as one's 'significant other', to the extent that one might identify more strongly with that other than with oneself:
"Lolita, light of my life, fire of my loins. My sin, my soul."
Whether this particular quotation from Vladimir Nabokov's Lolita can be reasonably asserted to mean that is debateable, I just felt like posting it as an example.
Examples don't have to come from the real world, literature is full of them, and don't be afraid to use fictional anecdotes as a form of evidence in your piece.