long before the politically correct speak of referring to everything in the Western Hemisphere as "American" became common.
However, someone from say Colombia might say they are "American," but are actually referencing the fact they are from the continent of South America
A large portion of my father's side of the family lives in Brazil, and I lived there as a child, and have visited other nations on the continent. For someone from South America to self identify as a "American" is almost unknown.
Then it is possible I have been misinformed, because I took a class that stated the opposite: that in many Latin American countries in particular, in Both North and South America, many people identify themselves as "American," regardless if they are Mexican, Guatemalan, Panamanian, etc. Perhaps it is more a Central American thing, then? Or maybe it is a small but vocal amount of people who do this? Or it could possibly be something more akin to a train of thought rooted mostly in academia? Or maybe a little bit of all, where maybe it is a very small minority of people in said locations that practice this, and academia has picked up on this and exaggerated the extent of the practice, making it sound more widespread than it really is?
Anyway, my understanding for the rationale of people in the Americas outside the US calling themselves "American," is somewhat of a reaction to the fact the US citizens referring to themselves as American, and they feel the term has been highjacked by the US, and minimizes or excludes other people that live in the Americas, but outside the US.
And, BTW, this is not to be confused with indigenous peoples of the Americas (including Native Americans of the US and Americas in general, AKA Amerinds), whom also refer to themselves as American (which one has to agree that they are in fact the ORIGINAL Americans), regardless if they are in the US or elsewhere in North or South America.