No, I mean that they were the first independent country in the Americas (by which I mean the first independent country recognized by Europe, of course). If Mexico or Canada or Brazil had secured their independence before the Thirteen Colonies, they might have become known as “America” instead.
Technically, Mexico’s name is “United Mexican States” Estados Unidos Mexicanos. Everyone refers to them as Mexico and the citizens as Mexicans. Just saying.
I get why this confuses some people, but what else would we call ourselves? United Statesian? People from the US? All very awkward. Plus that would rip off other countries with “United States” in the title. We can’t win. America, rah!
I think anyone getting upset because Americans call themselves Americans really needs to find something more important to get upset about. The name of the country is the United States of America, shortening it to ‘America’ is hardly controversial. Nobody is confused.
Well, there’s no such country. 
Canada’s entire official long form name is “Canada.” You won’t find “Dominion of Canada” anywhere; that’s a common misconception. It was used a long time ago but no more, and the Constitution officially dubs the country “Canada.”
I watch a lot of British panel shows, and hear “the States” fairly frequently. But I never hear y’all call the inhabitants anything but “Americans”, or very occasionally “Yanks”.
Parenthetically, I’ve read that “septic” is a nickname for us, based on rhyming slang: “septic tank = Yank”. But I’ve never heard it used. Is that actually a thing?
DPRK is right on, but would there be a viable alternative?
“United Statesians”? “U.S. A-ians”?
Brit born and raised, and I lived in London for a few years, and IME rhyming slang is almost entirely a guy ritchie movie thing at this point.
One or two words have stuck: e.g. butcher’s, porkies. But can’t say I’ve ever heard “septic”.
USAmians.
MUSAGA!
There is, contemporaneously, a slow and gradual but real development going on along similar lines in Europe: “European” is increasingly (though still not exclusively) understood to refer to the EU alone, as opposed to the entire continent.
I believe the term is “seppo”, and I’ve heard it verified by an Aussie (somewhat sheepishly). IMHO, it’s more an insulting term of endearment than a serious insult.
It’s “Seppos,” which is derived from “septic tank,” but no one says “septics,” so far as I know. I believe that this is a somewhat old-fashioned term, though, having first become popular during World War II.
There was an obnoxious Doper who insisted on calling Americans “USers”.
Breaking news (actually, it dates from May but conservative news outlets have just found out and are having lots of fun with it): a Stanford Elimination of Harmful Language Initiative discourages use of the term “American” in its usual context.
“This term often refers to people from the United States only, thereby insinuating that the US is the most important country in the Americas (which is actually made up of 42 countries).”
Once your eyes stop rolling, you may notice that Stanford’s preferred replacement term, “US Citizen” is even more problematic, as it insinuates that you’re not an American unless you’re a citizen. Tough luck, recent immigrants (legal and otherwise).
The FINAL decision: “U.S. Inhabitant”
I have spoken! LOL
Even referring to America as “United States” or U.S. has an unfortunate air of geo-centrism, as there are other countries which have had “United States” in their name.
The current official name of Mexico is “United Mexican States”.
Maybe USers should just call it “The Country Formerly Known As America”.
What’s the matter with the traditional (on the internet, that is) name of Merkins?
Nothing, if you pretend that you don’t know that a “merkin” is a pubic wig.
That’s a feature, not a bug.
Not sure how widespread it is/was but citizens of Hawaii are sometimes annoyed at tourists talking about “returning to The States”.
That’s anti-non-contiguous prejudice.
Gotta go with the classics. We are the “We-uns,” because we refer to ourselves as “US.” We are still having trouble with the successor state to our mortal enemy the “More-Wes.”