"American" is the word in English. Most of the world, inclusive of all native English speaking countries, see the American continent as split between North and South America, so technically when they reference America, they aren't referencing the entire continent because that isn't a concept that exists for them. "Americas" is probably the closest term we have for the wholistic continent, but rarely would someone say "this person from the Americas".

"American" in the Spanish form is very different. Spain has seen America as wholistic, largely because of imperialistic and colonial motivations, so when they say "American", they referencing a person from the whole continent. So naturally, when people from countries formerly colonized by Spain grow up learning Spanish, they're also going to have the same perspective. Contrary to popular belief, it is not a concept that was founded by indigenous groups within the Americas, but by Spain.

I don't think people realize this, but arguing against the concept of "American" signifying as "a person from the United States" only further supports Spain's original definition of "America" which is founded off of colonization. It's ironic at best.