As a matter of actual geological fact, North America and South America are separate continents (they have their own cratonic cores). Geographically they are considered distinct in the USA also. Central America is a cultural or political region refering to the isthmus -- it is not a continent in any sense.
Nationals of the USA are called "Americans" by USA nationals, and by people from other parts of the world, including Japan, Russia, etc (in their own phonologies). Canadians refer to USA nationals as Americans, and do not call themselves Americans.
Europeans frequently object to USA nationals calling themselves Americans, claiming that the word should refer people's of both North America and South America. People from South American nations seem to feel the same. Mexicans seem to me much more likely to refer to a USA national as "Americano" than they are "Estadounidense."
USA nationals will sometimes describe people from North and South American nations as be from "the Americas."
Many Europeans object to calling themselves Europeans.
Ex: From my travels and conversations - the English don't refer to themselves as Europeans. I asked them what continent they lived on. Doesn't matter.
It's hard to recall what I was taught about Central America, but I believe it was that it's sort of a region shared between both and only a colloquially separate entity.
(Unfortunately though, I think that ignorance is fairly common for a lot of aspects of life for people raised in the US.)
Side note: I was going to say "aspects of life for Americans", but realized Americans means more than just those in the US. So I propose a new term for "people from the US". USers. :)