Borough names (like
First of all, the second "New York" in "NY, NY" refers to the state of NY; the rationale for using that term should be obvious.
Regarding the first "New York"...
Remember, "Manhattan" is a borough name. Boroughs are citywide designations that were created relatively recently (in 1898).
Loooong before that, the same territory was called "New York County" (a state designation) and, of course, "New York City."
At one time (pre-1874) "New York City" consisted ONLY of Manhattan Island. The rest of the present-day boroughs were independent and autonomous towns, villages and cities. As NYC grew (in 1874, 1895, 1898) and shrunk (1899), the term "New York" -- and presumably, its postal nomenclature -- still clung to Manhattan.