There are many fine shades of political analysis. Oligarchy is a broad category, meaning rule by the few. Oligarchies include monarchies, aristocracies, plutocracies, and so on, and they can include timocracies, kleptocracies, etc.
I think the main question raised by the media's obfuscation on this issue of oligarchy is whether American democracy is still functional or even relevant. Democracy, rule by the many, should be the opposite of an oligarchy. But if democracy is self-sabotaging or if it's vulnerable to capitalistic countermeasures, democracy might become a sham, in which case the true government might be a shadow one.
The question is always who has the real political power in a society, the many or the few. That's the question that's being set aside by this journalistic cliche about "Russian oligarchs."