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.
I can count 4
Potentially any island is a continent if your cutoff is low enough
It turns out that there are various models for the number of continents, and that is (phew) known in Spanish, too. See the Wikipedia page [1] (link to Spanish version) for instance. This is for European Spanish though, but I couldn't find a version of the page in es-AR.
[1]: https://es.wikipedia.org/wiki/Continente#Modelos_continental...
You're definitely right about there not being a word for someone from that continent though.