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.