>>652967A monarchy is usually defined by the monarch
owning the state. Historical elective monarchies were typically a bunch of princes who owned their respective states deciding who would be the "head prince" and lead the rest in some capacity, which is what their real role regardless of whatever their official title was.
I don't believe the Pope or the Cardinals actually own the Vatican or the Church in any official capacity. I think he's styled as some kind of monarch, but in reality is just the high priest and supreme leader of a theocracy.