Except the original that you replaced with your copy. You're opting for an external functional description of identity but the discussion is about the individual.
It should be understand as conceded that a perfect copy of me would pass any Turing-style test applied by an external auditor that the copy is me, but that doesn't mean that I am the copy.
Its different in a sense, sure. But consider: if I replaced it so perfectly that it was atom-by-atom identical, then God himself would not be able to say if it was you or not. Unless we admit to some external agency that defines 'you' that is not present in the mechanism e.g. a soul.
It should be understand as conceded that a perfect copy of me would pass any Turing-style test applied by an external auditor that the copy is me, but that doesn't mean that I am the copy.