But it is part of the 5 boroughs even though geograhically it's here on Long Island. FWIW, I also consider Manhatten to be "The City" as well.
I also refer to Manhattan as "The City", if I'm going to Brookyln or Queens, I say "Brooklyn" or "Queens". I don't feel like I'm in "The City" unless I go over the bridge or through the tunnel!