Answer:
No, not exactly. The five boroughs of New York City are Manhattan, Queens, Brooklyn, The Bronx and Staten Island.
Each borough is also a county in state government: Manhattan is New York County, Brooklyn is Kings County, Staten Island is Richmond County, and the other two have the same names (Queens County and Bronx County).
So, the terms "Manhattan" and "New York County" both refer to the same piece of land. But Manhattan is the name of the borough (and the island), while New York is the name of the county. There's no such thing as "Manhattan County."