They're comparing to NYC, not Manhattan. NYC includes 5 boroughs (and some of them are _large_), and while Manhattan and inner parts of Brooklyn, Bronx, and Queens are dense, most of Staten Island and outer parts of Bronx and Queens are relatively sparse.

You can use the same trick to come to the surprising conclusion that LA is denser than NYC. Population-weighed density is definitely a better way to measure density if you want numbers that reflect intuition about how dense a place "feels" rather than merely reflecting what arbitrary political boundaries were chosen