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Why are the United States often referred to as America?
Is it because there wasn't a proper adjective like "United Staterns" or something?
Why are Canadians not called Americans (or the rest of the inhabitants in the continent for that matter)?
Isn't it like calling France Europe and French people Europeans and implying that British people, Spaniards or Germans are not Europeans although they do live in "Europe" ?
I have always had this question.
Remember that the U.S. of A. is not the only United States in the world! Just across the Rio Grande is Estados Unidos Mexicanos, the United Mexican States. In a sense, then, the world's largest economy doesn't have any name at all that it can call its own, even on its own continent. But people have to be able to call us something, so "Americans" it is.
To go a little deeper, sovereign states without unique geographic identifiers in their names are unusual. Apart from your occasional North and South Koreas and the like, I can only think of a few others at the moment:
The "rule," then, seems to be that we use the geographic component of the name when doing so causes no confusion (and sometimes when it does). Otherwise, we use the most unique component of the name. As there is no other country in the world with "America" in its name, American qualifies on both counts.
I believe it's a combination of factors, mostly accidental. Since anything derived from United States (such as United Staters) would be awkward to say and not at all catchy, it's not a surprise that nothing of that form caught on. The full name of the country is the United States of America, and out of that, Americans certainly fits the bill as catchy and reasonably comfortable to say. After that, it's pretty much just tradition and inertia.
Why aren't Canadians and Mexicans also Americans? Well, again, mainly tradition and inertia, and the fact that those countries happen to have picked "proper" names for themselves. Also note that usually, North America and South America are treated as distinct continents, and citizens of Canada, Mexico, and the United States are all referred to as North Americans.
I think the main reason is that United Staters is really a mouthful, and neither of those words by themselves is very descriptive, whereas "America" a least narrows it down to a single hemisphere.
The other inhabitants of the Americas are of the opinion that we (in the USA) picked this term ourselves (which is debatable) and that it is emblematic of our self-centered and egotistical nature (they have a decent point there). So when speaking or writing in a forum where there are liable to be folks from the western hemisphere outside the USA present (particularly people from Latin America) it is a good idea to avoid saying "Americans" to mean USA residents. I have seen people angrily set straight for doing that.
As an alternative, the Brits like to use "Yanks". I don't have a big problem with that, but many southerners wouldn't like it, as that's close to the perjorative term they use for the northerners that beat them in the US Civil War. As mentioned in the comments, you often see USAsians online these days. It's clever and I like it for written works. However, when spoken it sounds so much like "Asians" that it could be confusing.
I used to think that denoting citizens of the United States of America simply as "Americans" was unwarranted on both geographic and ethnic accounts, unduely squatting what should be reserved as a denominator for all inhabitants of the Americas to the benefit of US residents. And it probably is.
However, this viewpoint is a short-sighted moral one, and doesn't take history into account, which is full of convolutions, and not always guided by evil intents, but sometimes just by political, cultural, or economic gravity.
So to answer the question as to why we think of the United States when we say "Amerikaner", "Américains", "americani" - I think it's simply an expression of the importance the United States have come to occupy in the world during the 20th century.
When something is important, people tend to refer to it a lot, and they don't always use the formal terms, especially when they are a bit lengthy and pompous and don't fit patterns like "the Germans, the French, the Italians, the English", "die Deutschen, die Franzosen, die Italiener, die Engländer", etc - there simply had to be "the Americans" just to align with the other denominations for people of various countries.
It couldn't have been otherwise because in the phrase "the United States of America", or "the American Union", the union and the state are anaemic political terms, while America is about the land, which is real and palpable, and lends blood and life to the term.
Consider that the song is called "America the Beautiful", not "Union the Beautiful".
So, to sum it up, there needed to be a term in alignment with the established terms for other, granted, more traditional and ethnically more homogeneous nations; and America was the only serious pretender to form the root of that term, the others being too abstract and lifeless.
The fact that non-US Americans are sort of disregarded by that linguistic choice merely goes to show that their countries have not, like it or not, risen to such a dominant role in the world.
