How does something change and yet stay the same?
If you rip a corner off a sheet of paper, how do we recognize it as the same sheet of paper? What defines anything as "that thing," if at any point anything about it can change completely?
A man with a finger cut off is no longer the same set of atoms as before, but he's still the same man.
I would say that there is one or more "defining characteristics" of any one object. In the example of humans, I would have to say that a person is defined as a "unique person" by their past--which is the one thing about themselves that they cannot change.