Since we have North America and South America, the term "American" could be interpreted as referring to anything within either continent. Yet most of the time when the word "American" or "murica!" is used it is referring specifically to the United States.
e.g.
I speak American, yet clearly that is not the only language in the Americas.
"America the Beautiful" is referring to the US primarily.
"American-made" not "Made in Mexico"
Yet isn't Mexico a part of America?
So my question is really about the origin of the term "America" and why it used in the manner today to refer to the United States.
In English you use the full name of the continent in the continental demonym. So the Continental demonyms are "South American" and "North American" in English because the English world classifies the Americas as two continents. So no, "American" cannot mean anything from either continent in English.
English also tends to use the most unique identifying word in the name of a country to form its demonym. For "The United States of America" that word is America so the demonym is "American". For "The United Mexican States" that word is Mexican so that is the demonym.
Also the first country to claim American as their demonym was the USA, and it was also used by Europe for us before a single country left their respective empires.