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> As for "a British accent". Would an American say there's "an American accent"?

Of course, just the same as British. "American accent" for foreigners usually means "Southern Foghorn Leghorn" e.g. Daniel Craig in Knives Out, but can also mean New York, Boston, Chicago, Dakota (aka Fargo) or some other variation. To Americans it really means "unaccented" Midwestern English, aka broadcast (radio or TV) voice.

Americans know "American accent" from "British accent" by the same standard, the British "unaccented" voice is called RP and sounds posh or overly fancy to most Brits, but Americans don't distinguish RP from Cockney, Yorkie, Jordie or any of the other variants. RP is the broadcast voice.

To be clear: I used to be the same. "British" was one accent that Michael Caine, the Queen and Chris Ramsay all spoke. (For Americans: "Robert DeNiro, Harrison Ford and Dolly Parton have one accent" is equally wrong and hilarious.)