I live in Brooklyn, I work in New York City, both for the full year. Do I have to pay New York City individual income taxes?

The tax you are referring to is the NYC Resident Tax. And yes, you have to pay it since you are are a resident of New York City. It is calculated as part of your NY State income tax return.

Last time I looked Brooklyn was a borough of NYC. Therefore, you both live AND work in NYC and are subject to the NYC Resident Tax.