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Douglas_Knight comments on Shock Level 5: Big Worlds and Modal Realism - Less Wrong
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It does not specify an probability for each point, but a density, which is only turned into an probability by integrating. The probability of being at a particular point is zero. More strongly, the system has countably many qubits.
Sure.
So we agree: far from there being 'only finitely many copies', if there's any sense at all to be made of the 'number of copies' then it is infinite and any (or 'nearly any') possible configuration of the matter making up your body gets some non-zero density or probability depending on whether by configuration we mean a 'point' in phase space (if that's the right term...) or a little 'cube'.
Your original post is correct. I think "continuum-many" is misleading, but I can't object much given the context. I don't remember what I was thinking.
ETA: I would think of an eigenbasis and say that there are countably many electrons, none of which is localized.