Yes, "unique factorisation" is a nice argument yet it seems to say more about the human mind (which finds uniqueness to be elegant or at least easier to think about) than it does math (which has no qualms about infinite sets).

This is beginning to remind me of the arctan and inverse abs threads in which someone vehemently defended the arbitrary convention of picking a particular single value as the only "correct" solution of a multivalued mathematical relation.