It only becomes an -archy when the actual literal power is openly and legally held
Otherwise, neckbeards would argue that literally every society in history has been a kleptocracy, which is a far more apt description of the sarcastic viewpoint on this society than "oligarchy," which is when rich people are put into office which matches the thing they are good at, would be
Oligarchy does not mean "rich people hold power because they're powerful," or anything even slightly like that
Think about the society that you see in comic book alternate universes every so often, where they make Thomas Edison the chief of industry, et cetera. The nonsense one where we somehow have the ability to divine who's good at what by looking at the history of their wallet, and on that basis make meritocratic elections. In such a society, a lottery winner even who exceeded the wealth of others would wield nonetheless no power, because it wasn't earned.
Practically speaking, there's no way such a thing could plausibly exist. But that's what the word actually means. It's one of those things where people trying to use it to sound deep and wise end up undercutting themselves (like "utopia.")
Perhaps people are bored with the word billionaire and want to use an edgier sounding synonym. That is how languages work - they evolve. (i.e. the word literally is now mostly a synonym for figuratively.)