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Mail chutes were a common feature of buildings for many years, and not only in New York City.

The sixth (1970) and seventh (1981) editions of Architectural Graphic Standards discuss them on the “Planning for Postal Service in Office Buildings” pages. From the latter:

  CHUTES: Used in buildings of at least four stories.
  The chute must be approximately 2 x 8 in. in cross
  section and extend in a continuously vertical line
  from the beginning point to the receiving box or
  mailroom. The interior of the chute must be access-
  ible throughout its entire length. Chutes installed
  in pairs are constructed with a divider and dual
  receiving boxes. Chutes are for first class mail only.