If a mental and physical clone of a convicted serial killer could be magically made, the justice system would have a dilemma on its hands. Should the new copy be punished, too, even though that body never actually committed the crimes? (It carries the same personality and memories of the one that did the deed, but is potentially free to roam at large in the world.)
In terms of living things, the persistence of configuration or form is apparently what matters to us, rather than the specific components constituting _X_. Except when it doesn't -- identical twins and thought experiments like the above.
There is also our special emphasis on brain configuration, in contrast to the rest of the human form. If a person suffers total amnesia and acquires new memories and tendencies, then we don't consider them the same "person" -- albeit legally they're treated as the same individual, and in terms of moral duty they're still filling the slot of _X_ family member. Deep mental illness sometimes renders similar results. ("
Drew is a totally different person since s/he went mad.")
But even our structure is always changing at the microscopic level, especially the neural "wiring" or relational connections (our psychological template). That seems to be okay in terms of identity as long as it's a slow and incremental process.
In a
flip book, the scene on each page is different or separate from the others (a distinct entity), even though they often repeat the same slightly altered configurations (contents) in order to produce the illusion of animated figures. That distinctness is also the case with all versions of time -- even presentism, since the last configuration of the universe is replaced or eradicated by the next universal "now" or present moment/state. They just don't co-exist in presentism, as with the pages of a flip book or with the frames of an old-fashioned motion-picture filmstrip.
So in a sense, it's really just the replication or survival of personal memory in each different "snapshot" that convinces us that each parallel, modified version of ourselves (in a sequence) is literally the same person. That along with similar information storage in the environment, which likewise enforces the idea that the same individual endures.
Horowitz, Arshansky, & Elitzur: It seems that Einstein's view of the life of an individual was as follows. If the difference between past, present, and the future is an illusion, i.e., the four-dimensional spacetime is a 'block Universe' without motion or change, then each individual is a collection of a myriad of selves, distributed along his history, each occurrence persisting on the world line, experiencing indefinitely the particular event of that moment.
Each of these momentary persons, according to our experience, would possess memory of the previous ones, and would therefore believe himself identical with them; yet they would all exist separately, as single pictures in a film. Placing the past, present, and future on the same footing this way, destroys the notion of the unity of the self, rendering it a mere illusion as well. --On the Two Aspects of Time: The Distinction and Its Implications (1988).
But such a so-called "identity problem" is really no different with respect to the other "simply depicted" spacetime options, and presentism's lack of spacetime.
For instance, the "Growing Block Universe" is retaining (instead of destroying) each new created state of what would otherwise be the presentism process.
And the "Shrinking Block Universe" is doing the reverse of moment by moment destroying a future that already exists.
Whereas the ordinary Block Universe (above) simply eliminates the constant creation/destruction of presentism, and the exclusive constant creation or constant destruction processes of the other two. (All versions of "you" have a privileged or equal existential status, rather than an ephemeral or mitigated one).