Then I guess I won’t bring up “septics” around you 
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But as far as who’s calling us that, I believe it’s us. That is, it’s based on the 'Merkin way of speaking, as different from those furriners. It’s pretty much are own petard we’re being hoisted upon.
To return to the OP, our idea of “continents” started with the Greeks, who recognized three continents, Europe, Asia, and Africa. They didn’t have any idea how big those continents were, or how they connected, they just knew that the Mediterranean and Black Seas separated Europe from Asia, the Red Sea separated Asia from Africa, and the Medierranean separated Africa from Europe.
The only reason they even came up with this idea of continents is that they lived at the conjuction of three of them, and the main way to go between them was by water. To even come up with the idea of continents you have to know of more than one, otherwise you just have the idea of “land” and “sea”. So saying “All known land is part of a continent known as Lemuria” doesn’t make sense, “We live over here in Lemuria, those guys live over there in Furnituria” makes sense.
Since native americans didn’t have an “over there” to compare with “here”, there’s no need to have a name for “here” except perhaps to distinguish “land” from “sea”.
Now, now… If there’s a vowel change from o to a, shouldn’t it be the other way around:
Canadons
There have been a number of attempted alternatives, but none have stuck. I’m partial to Usian, myself.
Slight hijack but tangentially related: did any of the advanced Indian civilizations (Aztec, Mayan, Incan, etc.) have maps? That’s a concept you’d think would have presented itself and come in handy (though I’m sure they’d have been geographical rather than political if they did).
I know that with the Mayans and Aztecs most of their written records were burned by the Spanish, but at least a few did survive in some form and have been discovered since.
As has been pointed out before - and I’m Canadian - there’s no continent called “America.” There’s a North America, though, and Canadians are North Americans. “America” in common usage refers to the United States just as much as it does anything else. English words are generally defined by usage and for 250, maybe 300 years, if you said someone was an American, most everyone in the English-speaking world understood you were referred to someone from what is, or is now, the United States of America.
On the other hand, there IS another United States… the Estados Unitos Mexicanos. So actually, “America” might be a more specific term than “United States.”
You know, I always forget that little fact. I’m happy to be enlightened, again.
But do the Mexicans ever shout: E.U.M.! E.U.M.! ?
But do we know for sure that American Indians do not or did not have a concept of “over there?” If the theory of migration over a narrow temporary land bridge is remotely true, then oral tradition might refer to an “over there” where distant ancestors came from. Any Dopers out there who know anything about such a possibility?
I think those that use the term are fully aware of its meaning and intend the waggery.
By the way, I believe “merkin” isn’t so specific. It can also be used by men and it can also be used to simulate chest hair.
Yes, but the land bridge wasn’t narrow, it was as wide as the entire state of Alaska. We’re not sure how fast the migration occured, but it probably took at least a few dozen generations. And even then the idea of continents as large land masses separated by water wouldn’t make much sense, because at that time Siberia and Alaska weren’t separated by water. You wouldn’t think you were on a new continent if you walked from China to Siberia, so why would you think you were on a new continent if you walked from Siberia to Alaska and from Alaska to Canada? There’s no indication that you’re on a new very large land mass.
Yes, Europe is connected to Asia, but the ancient Greeks didn’t really know that, since they didn’t have a good idea of how large the Black Sea was. It seems to me that only a seagoing culture could invent the idea of continents, large land masses divided from each other by seas. If you never sail from one continent to another there’s no reason to invent the concept.
Brazil is another example of a country that, at one time during its history, used the designation of United States. Nowdays, it’s the Federative Republic of Brazil.
Regarding the usage of “America,” I’d like to remind that there’s still the Organization of American States. And my impression from books, news, and travel in Latin America is that there is a certain sentiment among many Latin Americans that’s opposed to using “America” to describe the United States alone.
Many groups have oral traditions of having migrated from elsewhere to their present locations, but this does not mean they had any concept of having come from a different continent. (Often, the tradition involves coming up from an underground world). In any case, as Lemur866 says, the so-called Bering Strait “land bridge” was not narrow, but very wide, and would scarcely have been recognized as different from the adjacent continents when it was being used. Also, with increasing evidence for “pre-Clovis” migrations into the Americas it seems probable that some groups may have used coastal routes rather than the land bridge itself.
My personal experience (and it’s considerable) is that Latin Americans mainly do this to needle the gringos. 
They are not seriously interested in being referred to as “Americans” themselves. And as I said, as an alternative to americano for USians, they tend to use norteamericano, which is also inaccurate. IMO, they’re welcome to use whatever term they want to refer to USians in Spanish (although estadounidense is better than norteamericano - and gringo is much more common than either), but shouldn’t be trying to dictate the meaning of a word in English. And in English, the most common meaning of “American” is USian.
Has everyone fogotten the infamous “USers” from the **aldebaran **days? Well, maybe some things are better off forgotten. 