Reading this headline (as a Dutchman) my brain went „Amsterdam has its own airport?” before going „Oh right, Schiphol…”. (Compare this to a New Yorker reading about New York Airport instead of JFK.)
Schiphol's IATA code is AMS, therefore it's the Amsterdam airport. :)
It also officially named Amsterdam Airport Schiphol, has the words "Amsterdam Airport" in its logo, and appears as Amsterdam (AMS) in flight searches. It's not like there are any real alternatives for flying into Amsterdam
Thing is, NYC has three airports (JFK, LaGuardia and Newark Liberty), whereas Amsterdam only really has Schiphol.
Oh the confusion doesn't lie there; ask anyone about an airport near Amsterdam and they will think of AMS. it's just that for most Dutch, Schiphol (AMS) is a national airport rather than just Amsterdam's airport. So 'Amsterdam Airport Schiphol' sounds pretentious but correct, whereas 'Amsterdam Airport' just doesn't click — like adverbs without a noun.