Identical is not the same is indistinguishable. Two electrons can be perfectly identical, but still distinguishable in hidden-variable theories for example, because there their position and velocity are precisely determined at each point in time.
EM
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It’s annoying to read and doesn’t seem to add much to the point. It could be a whole lot shorter.
You assume that the wave-functions are ontic, or close enough. In hidden-variable theories of QM the particles have a definite position and velocity at each point in time, and I don’t see a problem with adding a hidden variable each particle that makes them nonidentical, except that this would violate occam’s razor, but that is a probabilistic argument, not an absolute. And while hidden-variable theories are not falsifiable, you could have “slightly-hidden-variable” theories, which would need to be ruled out by experiements.
You could argue that the added hidden variable is not part of the particle, but then your particles would be identical by definition, which makes this whole discussion moot.