Answer:
The City of New York (comprising all five counties or "boroughs") is referred to as New York, NY. The island and borough of Manhattan is New York, NY.
When you write a letter to someone in Manhattan, you write "New York, NY" in the address. If you are writing to someone who lives in one of the other boroughs, you write, "Brooklyn, NY" or "Bronx, NY" or "Staten Island, NY" -- except for Queens. Queens is the only borough where the neighborhoods retains their status as a "town" of sorts. If you live in Astoria, Queens, you would write "Astoria, NY".
However, when writing or speaking other than for a postal address, you can use the term "New York City" and be equally correct and understood as if you wrote or said "New York, New York."
You can say "New York City" or "New York, NY", but it is not usual to say "New York City, NY"