Just to throw out a few references, since they seem unmentioned so far: The Ship of Theseus is an ancient recognition of a similar problem. Greg Egan has writ

Just to throw out a few references, since they seem unmentioned so far: The Ship of Theseus is an ancient recognition of a similar problem. Greg Egan has written two stories that deal with copy-vs-continuity of consciousness: Diaspora and Permutation City. Both start with the premise "what if we could digitize our mind?" but go in different directions.