There are multiple, separate issues in the question.
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."
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."