As a Brit living in Canada when I say American I mean US American, not North American.
I've never met a single Canadian in my ten years here who has ever referred to themselves as anything other than Canadian, in the same way Brits rarely seem to refer to themselves as European.
I never say I'm European, I say I'm British or English.
Although the subject of being American has never beee raised that I can recall, I can't imagine anyone I know wanting to be called American rather than Canadian.
In fact, people tend to want to emphasize the fact they are Canadian, in my experience.