Governments don't have ethnicities so one would think that it's correct to say 'the Russia/US government'. However, I countless times heard people say 'the Russian/American government'. Which one is correct?
When you use nouns as modifiers, this is called using them attributively.
X Y attributively associates X attributes of X with Y. The association is typically not strong enough to say X is origin/creator of Y.
Chicken soup - soup has chicken, but a chicken didn't create it..
Face mask - the mask is for your face, but your face didn't create it.
The adjective form of nationality words specifically means "originated from that country/culture/nation". So you want to use those when you want to say that.
It's not about ethnicity. "Russia" is a noun, so it can't be used to modify another word. "Russian" is the adjective form, which can be used to modify words like "people" or "government", so you can say:
The Russian government
The government of Russia
but the following is wrong:
The Russia government (wrong)
..because it puts two nouns right next to each other without anything to connect them.
(It should be noted here that "United States"/"US" is a bit unusual in that the noun form is "United States" and the adjective form is also "United States" (they're the same). This isn't true of most countries' names, though.)