This blows my mind, because it never occurred to me that this would actually be taught differently. But yes, afaik, in NA we're taught that North and South America are two separate continents. In Canada however, Americans are from the USA, but we don't generally refer to the USA as America. Only they do that. We refer to the middle nation in NA as The U.S.

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.