A sociology professor once informed his students that many people from countries in South America and our Northern American neighbors in Mexico and Canada consider our citizens from the United States arrogant because of calling ourselves, Americans. Their discontent with this proclamation might make perfect sense because they too live in the "Americas"; both South and North America; the latter continent of which we hold only a portion of its land.
Yet to them from these other nations, we would indulge them to understand; not because we attempt to claim exclusive privilege to all the land therein and their continental name, but to an ideology started long ago, which we have accepted as our own.
When the forefathers of our nation in North America labored intensely to establish our government, perhaps they did not exercise as much effort to contrive our country's name. They decided to call our nation the United States of America. The name does indicate a union between states, yet our forefathers may not have considered how our citizens should and would be called. To call oneself, the United States of American, is elongated and awkward. If we tried to simplify this name by use of the acronym from the letters and add an "n" at the end in order to denote association to this nation, should we call ourselves "USAN"s'? This wouldn't make any sense and does not sound very good; like something someone would throw up.
So with our dilemma for something to be named, it is easier to call ourselves, Americans.
Also embedded in that name we call ourselves are subdivisions derived from its root name: Native American, White American, Irish American, Black American, Italian American, Mexican American, Jewish American, Chinese American, Arab American, Japanese American, German American, etc., etc. Each of these groups distinguishes the variety of races and cultures which make up the heart of our nation. And each of these have their histories and stories intermingled with the others of how they have shed blood to live here and have rights to claim the nation as its own. So when the national anthem sounds, most of us will stand in respect for what most of us care about, being an American.