The Exchequer

[I spent the winter writing songs about getting better]

Is the term "Exchequer", in the UK the government department that is responsible for collecting taxes and paying out public, also used in the US?

No.

If not, what is the American equivalent to the Exchequer?

That duty is handled by the Internal Revenue Service.

[I spent the winter writing songs about getting better]

If not, what is the American equivalent to the Exchequer?

That duty is handled by the Internal Revenue Service.

Is it correct/common to use the word "revenue" as a metonym for "Internal Revenue Service"?

For instance:

"Surrey contributed more income tax to the revenue between 2009 and 2011 than anywhere apart from the City of London, the county council has said."

"They do not pay tax and cost the revenue hundreds of millions of pounds a year in what petrol retailers are saying is a 'crime wave'."

"The son of a man killed on Bloody Sunday has said 'the revenue got off quite light' when it spent £200m on the Saville Inquiry."

I'm asking because two different dictionaries suggest this translation for the word I want to translate into English, and it simply sounds strange to me. 🤷🏻‍♂️

No, the Internal Revenue Service is simply called the IRS in shorthand.

None of your sentences make much sense with "IRS" in them though, the IRS is the agency responsible only for collecting taxes and paying out refunds, not all government spending. The American equivalent of what you're referring to would be the Department of Treasury, of which the IRS is a sub-bureau of. "Treasury" would be the most accurate word to use in those sentences in an American context.

[I spent the winter writing songs about getting better]

"Treasury" would be the most accurate word to use in those sentences in an American context.

That's good cause that was also one of the choices in the dictionaries.
"Revenue department/office/board" would be accurate translations, too, I guess?

Those would be accurate if referring to state or municipal equivalents. It's kind of awkward federally. But most state equivalent agencies are called the "*state name* Department of Revenue."

[I spent the winter writing songs about getting better]

Okay. Thanks for teaching me some English!
I think "treasury" seems to be the most apt translation.
It's a shame that the Latin word fiscus (literally "money basket") doesn't exist in the English language.

No doubt the source of the word "fiscal" though.