In terms of identity that is correct, but don't forget that when we refer to a particular person we are really using a label to point at a specific continuity of experience and agency through time, and unless qualified somehow (twelve-year-old Bob, Alice last year, you committed after yesterday's meeting, etc.) the default time point is either at the moment of reference or a moment just prior (the rest of the 4D 'worm' is incorporated transitively), and (again, by default) the 'now' pointer is thereafter updated continuously (at least in casual conversation in natural languages such as English).

It is pretty rare for continuity of experience (and agency) to be broken, and outside of science fiction physical continuity never is, so it isn't too surprising that English (and other natural languages I'm familiar with) largely elide the point-in-time qualifier by default.

We - both individually and in groups - are all of us Ships of Theseus, passing in the night.