Apart from names which had a hotel or car rental brand, etc., these are UDRP dispute outcomes involving domain names with "airport" in them:
The type of name you are talking about is
descriptive, not generic. The term "Anytown Airport" is not a generic term for airports. It is a geographically descriptive term for an airport in Anytown.
As with any descriptive term, the answer is going to turn on whether that term has acquired distinctiveness as a mark through longstanding and substantially exclusive use such that the public recognizes it as a mark. In other words, while a descriptive term like "American Airlines" starts out as merely descriptive, one particular airline has been that since 1939, such that people do not perceive "American Airlines" as a class of airlines originating in America. Instead, it is perceived as a distinctive reference to one airline in particular.
This is why your question has no single answer. As much as people like to believe that these things boil down to black-and-white formulas, the world just doesn't operate that way.
So, take something like New York City. If you say "New York City Airport", you might be referring to La Guardia, JFK, or even some of the other regional airports in the greater NYC area. It does not identify a particular airport.
A "Paris airport" could be CDG, Orly, or some other places.
A "Chicago airport" might be O'Hare, Midway, etc., although I think they are shutting Midway down.
So, all three of those would be merely descriptive phrase that refer to airports in those locations.
Now, there are some places which only have one airport, but which call their airport something else. If you are going to the "Orange County Airport" in California, then you would be going to the "John Wayne Airport" since they don't call it the "Orange County Airport".
But, as indicated in the chart above, there are places which only have one airport, have had it for a long time, and everyone knows it as the "(Whatever) Airport". That is the type of situation where you are likely to run into problems.
As a general proposition, this is just a subset of "descriptive phrases" generally. So, there are going to be differences between, say, a "convenient rag" or a "handy wipe", both of which mean more or less the same thing, but one of which is a well-known mark in the US.