well... I know many people in the US will refer to people as "European", or "African", "South American" or "Asian" not necessarily as natives of the individual countries the person is from. Some of the more... well known lets say (I'm not sure the right classification, maybe 'popular with Americans as vacation destination') countries like France, Italy, Spain, England and Ireland might be mentioned specifically but someone from say Hungary, or Montenegro, or Belarus? Most Americans I think would just describe that person as being "European".
So in that light I see no reason why Canadians and Mexicans wouldn't be considered North Americans, or even just plain Americans.
I'm curious, would you refer to someone from a country in Europe or Asia as "Eurasian"? To me, using "American" for anyone from North or South America is analogous to using "Eurasian" for anyone from Europe or Asia. "Eurasian" is a valid word, and I'm sure there are some contexts where it is useful, but I don't recall ever
hearing it used in a normal conversation. I hear "European" and "Asian" all the time, as well as words for inhabitants of individual European and Asian countries, but never "Eurasian." I can't think of a time I needed to refer to the combined population of Europe and Asia, but if I did, I would say "Europeans and Asians."
Similarly, "North American" and "South American" make perfect sense to me as ways to refer to the populations of those continents. But I can't really think of a reason I'd need a single term to encompass everyone in the Americas*. Once the term encompasses the inhabitants of an entire hemisphere, it seems like it is so general that it ceases to be useful. I would say "North and South Americans" if I needed to refer to that population.
*As katycoo touched on, I don't consider North and South America to be a unified landmass of "America." The two continents would be "the Americas" plural. Likewise, as a South Carolinian, I refer to North and South Carolina collectively as "the Carolinas" (plural), not "Carolina" (singular)**. It's often convenient to drop a few syllables when referring to both of them, but IME they are referred in plural, because they are a set of two entities with similar names rather than a unified whole.
**With an exemption for the song "Carolina in the Morning" due to poetic license
. (Sorry about the footnotes on footnotes--I seem to be channeling a less-funny Terry Pratchett today.