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United States of America = America United Kingdom of Great Britain and Northern Ireland = Britain

I'd say "Britain" more commonly refers to Great Britain, not the UK (i.e. Britain == [England, Scotland, Wales, *islands]).

Speaking from a US perspective again, for me "the UK", "England", and "Britain" are all completely synonymous with each other.

He knows how it came into being. I think his question was more along the lines of "should it?"