You are correct. NYC is composed of five boroughs, Manhattan (New York County), Brooklyn (Kings County), Queens (Queens County), The Bronx (Bronx County), and Staten Island (Richmond County). Manhattan is the densest, Brooklyn is the most populated, Queens is the largest by land area, The Bronx is the only one connected to the mainland, and Staten Island is the most suburban feeling one. NYC municipal government is kind of weird, because the city is comprised of five counties. Most city services are citywide (police, fire, education, etc), but each borough elects its own DA and Borough President (from what I understand, as a life-long NYCer, this position doesn’t have a lot of power and is often a stepping stone to become mayor). The Mayor and City Council represent the whole city, not each borough individually.

All of that being said, when most people think of NYC, they think of Manhattan, which is the “core” of the city. This (not very beautiful) infographic seems to be cherry-picking individual parts of cities to compare, so that’s probably why they chose Manhattan.