Some Americans think the "democracy" vs "republic" distinction is extremely important, and that "democracy" means something like "tyranny of the majority", hence why it is good that America is not a _democracy_, but a _republic_ or a _democratic republic_.

Some other Americans (there is some overlap) also think that the US is so large and diverse that essentially its States are their own countries and the US is more like its own continent, and talking about the American _nation_ or even _country_ is meaningless. It is a union of States (though it is rare that someone argues that the US is not a country).

I think they're making a USA vs America (continent) distinction.