"(I wonder if it sort of works the other way round, for him? if the parents for all his reincarnations are picked based on similarities with him, since he'll still inevitably be himself even when he reincarnates?)"

Yeah, it kinda has to - if they were too different, they would be very confused where this baby even come from. xD At least similar looks are necessary, if not personality. But they're still only similar, he can't have the same parents for every reincarnation (it would very quickly snowball into 'nearly everyone comes back as the same person' if you consider all relatives/ancestors, and that's definitely not the case), so in the end he doesn't actually take anything from them genetically.