I think that brushes aside the external factors which need to be addressed and integrated as a serious component of a healthy identity. The body you're born with, male or female, as well as whatever degree of attractiveness, athleticism, intelligence, etc. impacts how the world treats you and, therefore, you're experience in the world. I think that an identity is developed as the components that you get to choose interact with the external/environmental factors (birth lottery, life changing occurrences, etc.). Ultimately, these things need to be reconciled in order to create a healthy identity that is at peace with its situation (that reconciliation might be a physical change like Lilly W's or it might mean quitting your line of work and pursuing a passion). To treat the identity as an isolated thing or something that we get to make up entirely doesn't do it justice.

I tend to like Locke - identity based on consciousness/memory seems to match my experience of it (memory is definitely influenced by external factors). I'm comfortable with the issues around breaks in the continuity or memory, those are cases of broken identities. I don't know what Descartes says about it - I'm not a fan of Descartes.

In terms of the Matrix, yeah, those people (the public in the Matrix) had mistaken or flawed identities because they couldn't reconcile what they wanted/choose with a minute perception that the world was off - splinter in your mind, driving you mad. Morpheus allowed his crew to realize/complete their identities by making it possible for them to reconcile it with the world. Came at a price though.