Funnily enough, this is something I realized and decided to come point out to the thread before all the debate about Ozpin, but I guess that just works better for being more relevant. Namely, I'm not really sure anymore that Ozpin's immortality does work the same way it does in canon.
When I first read this I just accepted it as, "Oh yeah, she's mistaking modern Ozpin for her recent research assistant Ozma, and expects him to help her. Boy is she in for an unpleasant surprise."
But now I realize that wouldn't make any sense with the soul-joins-with-a-different-person-style immortality, because then Ozma and Ozpin would be completely different men and wouldn't physically look like the same person at all (as in the show). So the fact that Athame mistook Ozpin for Ozma means that his body is still the same from the beginning, which must mean he's a more traditional style of immortal in this AU right? No reincarnation, no possession.