Weird fun fact (as an Argentinian who went to school in England for a few years): in English-speaking countries, America is not a continent in the same way as in Spanish. In English they have two continents: South America and North America.

So the word "American" in English does not mean the same as "Americano" in Spanish.

There's really no natural word in English to refer to someone from "El continente Americano", because no such continent exists in English. That's why they use the word "American" to refer to someone from USA exclusively.

"American" to refer to USA exclusively does make sense either way because USA shares the continent with at least two other countries no matter how you slice it.

Frankly, the model with the single America continent doesn’t make any sense, because south and north Americas are so different in both geographical and cultural/historical sense.