New York City consists of five boroughs. The mayor of NYC is the mayor of all five boroughs. Each of the five boroughs also has an elected borough president, who serves as a sort of sub-mayor of that borough, but the city council and mayor set major objectives and budgets and stuff. How NYC dwellers self-identify their locale of residence is up to them.

https://www.nyc.gov/nyc-resources/about-the-city-of-new-york.page


P.S. NYC has a population of about 8.4 million people. That's about the same as the sum of the populations of these eight states: Alaska, Montana, New Mexico, North & South Dakota, Vermont, West Virginia, Wyoming.

I am from Brooklyn and that's what I tell people when asked. I have family in all the burrows plus Long Island and we typically refer to Manhattan as "the city" within family conversations or NYC to outsiders depending on the purpose of the conversation.

Manhattan = NYC

If you’re from one of the other boroughs, you’d identify yourself as such. Like, “My step-daughter goes to school in the Bronx.” From there, it’s only a short subway ride into “the city”.

NYC has five boroughs -- Manhattan, Brooklyn, Queens, the Bronx, and Staten Island. When one says "NYC", they usually mean Manhattan.