I think what underlies that 'obviously wrong' view in many cases is a recognition of the fact that in practice, we rely on continuity to establish identity.

A great many optical illusions and magic tricks depend on this: if entity A is here at T1 and entity B at T2, and I don't notice any T1/T2 discontinuity, I'm likely to behave as though the same entity had been here throughout.

Of course, generalizing from those intuitions to a more fundamental notion of some kind of epiphenomenal identity is unjustified, as you say.

Then again, claiming that what makes me who I am is functional activity or information content is problematic, also. It isn't clear, for example, that amnesia or brain damage makes me somebody else. Nor is it clear that if someone else is able to emulate me well enough to pass the equivalent of a Turing test, that developing that skill makes them me.

Mostly, I think is a composite notion, like . That is, we judge that identity is preserved by evaluating a close-enough match along many different axes, and there's no single property or set of properties that is both necessary and sufficient to establish identity.

I also don't think it matters very much.