There aren't very many borough's in the US (see here).
As you probably know, Brooklyn, New York is in New York, New York so the fact that the city data responds with New York is 100% accurate.
That said, I do see The Bronx in an sample call I made to the google maps api here:
https://maps.googleapis.com/maps/api/geocode/json?address=610+Waring+Ave,+Bronx+NY
[2] pry(main)> HTTParty.get('https://maps.googleapis.com/maps/api/geocode/json?address=610+Waring+Ave,+Bronx+NY')
=> {"results"=>
[{"address_components"=>
[{"long_name"=>"610", "short_name"=>"610", "types"=>["street_number"]},
{"long_name"=>"Waring Avenue", "short_name"=>"Waring Ave", "types"=>["route"]},
{"long_name"=>"Bronx",
"short_name"=>"Bronx",
"types"=>["sublocality_level_1", "sublocality", "political"]},
{"long_name"=>"Bronx County",
"short_name"=>"Bronx County",
"types"=>["administrative_area_level_2", "political"]},
{"long_name"=>"New York",
"short_name"=>"NY",
"types"=>["administrative_area_level_1", "political"]},
{"long_name"=>"United States", "short_name"=>"US", "types"=>["country", "political"]},
{"long_name"=>"10467", "short_name"=>"10467", "types"=>["postal_code"]}],
"formatted_address"=>"610 Waring Avenue, Bronx, NY 10467, USA",
"geometry"=>
{"location"=>{"lat"=>40.861204, "lng"=>-73.870415},
"location_type"=>"ROOFTOP",
"viewport"=>
{"northeast"=>{"lat"=>40.8625529802915, "lng"=>-73.8690660197085},
"southwest"=>{"lat"=>40.8598550197085, "lng"=>-73.8717639802915}}},
"types"=>["street_address"]}],
"status"=>"OK"}
You can see in this file that they're not currently supporting anything but those params. Maybe a pull request is in order as this feature would be good for me as well.