Yes, those references are for Manhattan only. The other boroughs have their own lingos.
Oh, yes. Sorry. They are different parts of Manhattan. Roughly, though there are specific (and to the north, different) streets bordering them, the UES, for example, is the area to the east of Central Park and the UWS, that to the west. It is 59th St. that is the southern border to all three, and this building is two blocks south, on 57th.
The next little chunk of the City to the south is ‘Midtown’, and on the eastern side of the island, it is Midtown East. Though there are also some smaller neighborhoods within those larger areas, and this building is also in a smaller neighborhood called Sutton Place.
Manhattan is only one of the five larger ‘boroughs’ of the City overall.