In that case, people with retrograde amnesia are already dead and we should just consider them a new legal person...

If you lose all your memories due to amnesia, you will still be you, experiencing reality as you, just with your "hard drive" being wiped.

Well some people who believe in reincarnation seem to also think that at least some memories do get preserved, perhaps subconsciously.

Then there is the whole dualist aspect: if reincarnation is real, something must carry over past death. Call it a soul, or an essence or personality or whatever. While I find dualism highly dubious, I can see its appeal.

That said, one could imagine, at least in principle, that through "some mechanism" your reincarnated body develops the same brain structure that you had when you died, perhaps with some differences needed to account for the different memories. If one entertains this fiction, I would argue there is significant similarity between the person who died and the person who they reincarnated as.