Yes but in the USA a "bin" usually refers to a generic category of containers, often rectangular. A "recycling bin" is a specific kind of bin, and it's almost always qualified as such. If you called it a "bin" out of context people would be confused or think you're trying to be British or something.

Yeah, I'd say it exists in a linguistic grey-zone where understanding is a lot more common than usage.

Practically no American ever calls a garbage can a "bin" (though like you say we do have a concept of generic 'bins') but a lot of Americans will immediately know what you mean if you say it, sort of like "flat" and "apartment" (nobody calls them flats in the US, but many people know what you're talking about if you say it).