I have lived in New York City most of my life. When some one, and I mean a resident refers to "The City" they refer to Manhattan even though five boroughs make up the entire city. It goes back to the days when Manhattan was the actual city.

Even the subway stations in the outer boroughs still often refer to trains going to "The City" or "Coney Island"

The old lines BMT (Brooklyn Manhattan Transit) IRT (Inter-borough Rapid Transit) IND (Independent Line) & Culver Line, were at one time private corporations.

It is just something we say without any further thought.

The same is true of Long island. We tend to think of Long Island as Nassau & Suffolk counties, however Long island is really Kings, Queens, Nassau & Suffolk counties, 120 miles long even though two counties are part of NYC, the road signs will not tell you this. Long Island City is in Queens county for example. Interesting - have a good day all.