So is Europe -> Eurozone? Is EU -> Eurozone? Is Schengen -> EU?

These are all quite clearly defined. Some (many?) may think that the EU is the same as Europe, but this is plainly false.

The EU is the EU, europe already has a definition, trying to conflate it with the European Union for ideological reasons is stupid.

I would never in my life read the word Europe and think of the EU first.

And when you hear 'America' you never think of the USA either?

The two are not equivalent. America is shorthand for the United States of. Mexico is short for Estados Unidos Mexicanos. Europe is a geographically defined continent.

If I were in Switzerland, Norway, Iceland, Serbia, Macedonia, or the Ukraine would it not be appropriate to say I was in Europe?

Well technically I think it depends on which side of Iceland you're standing. To the west you're on the North American plate and to the east it's the Eurasian plate.

Consider that the only country within the North/Central/South American land masses explicitly having America as part of its name is the United States of America. It makes a lot of sense really considering the global dominance of the USA relative to its neighbors that "America" is colloquially understood to refer to the USA, and not Canada, or Panama, or all of South America.