Why does Brooklyn have Kings County Hospital, not Brooklyn County Hospital?
Because Brooklyn is in Kings County. Another one is Staten Island which is in Richmond County and was the city of Richmond before it joined New York.
Because the borough of Brooklyn is Kings County, just like the borough of Manhattan is New York County and Staten Island is Richmond County. (The other boroughs are Bronx County and Queens County.)
New York has five boroughs which are also five counties in the State of New York. Only two of the five have county names that match the borough names. The borough/county names are:
Manhattan (New York County)
Brooklyn (Kings County)
Queens (Queens County)
Bronx (Bronx County)
Staten Island (Richmond County)
Yeah, I know ... it's really confusing!