Depending on where you grew up and were taught geography, America may or may not have been taught to you as a combined landmass from the Southern tip of Chile to the northern islands of Canada, or separate continents split near Central America.

There is no right or wrong way of defining that. It all depends on custom and convention.

The reason you say why people from the USA respond with the United States when people ask them where they're from is likely because it's a shortened version of the full country name. This is similar to asking someone born in the United Mexican States that they're from Mexico, or someone from the People's Republic of China that they're from China, or someone from the United Kingdom of Great Britain and Northern Ireland that they're British (or Scottish or Irish or Welsh), or someone from the Argentine Republic that they're Argentinian, or someone from the Boliviaran Republic of Venezuela that they're Venezuelan, or someone from the Republic of Korea that they're South Korean (although most people actually just refer to this country as Korea, but that might depend on regional differences too depending on which country you grew up in and were taught from).

Another reason might be how the USA's government is structured. We have a federation where the overall government is a sum total of Tribal, State, and Federal governments. People of indigenous tribes in the USA refer to themselves as Native Americans or Indigenous, while people from different states have names for themselves (e.g. Michiganders from Michigan, Californians from California, Kansans from Kansas, Hoosiers from Indiana). You might think that because the federal government, officially called the "United States" in our constitution, covers the entirely of the geography of the USA that that's how you would refer to people from that nation. And you would be somewhat right because the US takes on international relations per the duties outlined in the constitution. But it would be false to refer to the whole country as just the US. The whole country is the USA, and perhaps that is why people from that country refer to themselves as American.

Why can't we have a more nuanced discussion where we talk about how each country/culture prefers to be referred to? I think it's pretty asinine to refer to the people of, for instance, South Korea as South Koreans because that's my American conception of that country, when in reality people of the Republic of Korea refer to themselves as Hanguk-in or Hanguk-saram. I would be perfectly fine with referring to that people using that terminology.

Why do we have to force labels and categories onto peoples when we could just listen to them for what they prefer themselves