The western hemisphere is the Americas because it's composed of two continents, The North and South American Continents. IIRC from foreign language courses even in Latin American countries they are called 'The Americas'.

To me it's similar to the demonym 'New Yorker'. Is that someone from Buffalo, Schenectady, etc., New York or from the five boroughs or from the county of New York, ie, Manhattan?

What about Netherlander instead of Dutch since it's a corruption of Deutsch. We could go on about makes sense and what doesn't.