"North America" is the continent the USA is on, and South America is a neighboring continent. There is no continent called "America" alone.
https://en.wikipedia.org/wiki/Continent#Number
"Continent" isn't remotely as straightforward as many may think and the teaching on what and what isn't a continent (or counted as such) varies globally.
Different models are taught in different countries.
It feels weird to them because they're used to calling the Americas "America", the US "Estados Unidos" and Americans "estadounidenses". Well, it seems weird to me as a (United States of) American that they use those names. Our cultures are just different.
Calling the US "America" is no more "abnormal" than calling Americans norteamericanos (Even setting aside Mexico also being part of North America, what about Canadians?) or estadounidenses (given Mexico also being the "United Mexican States").