This kind of stuff is fun to think about. In my opinion identity is based on behaviour, not composition. So long as an entity matches a given behaviour profile, then for all intents and purposes that entity IS the thing that the behaviour is normally attributed to. Kind of like Duck Typing: "If it walks like a duck and it quacks like a duck, then it must be a duck."

So my response to your car example is that both vehicles are "the same car" since both get you around and both are owned by you.

In that sense, as your behaviour changes so does your identity. If you were able to 100% accurately encode all the actions you could ever take into a Markov Chain, then I would argue that Markov Chain is what makes you "you" at that point in time.