Oligarchs are, by definition, government.

I know it's a hip thing to say these days but it isn't accurate. I know everyone wants to one-up everything but they are just rich. Oligarchs are generally government officials who are given money and power by government.

I think people commonly use the word to refer to business oligarchs, which are not technically government:

"A business oligarch is generally a business magnate who controls sufficient resources to influence national politics."

https://en.wikipedia.org/wiki/Business_oligarch