Apple Thunderbolt 3 to Thunderbolt 2 and Thunderbolt Display I just bought this Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter  for my new MacBook Pro (2016) to connect it to my old Thunderbolt Display, but realized that none of my USB ports or Thunderbolt port work (so no daisy-chaining).

Is it something wrong with my setup or is this connectivity not supported by Apple's official Thunderbolt 3 (USB-C) to Thunderbolt 2 solution?

This does not sound like "to Thunderbolt 2" adapter to me, since it only supports multimedia.
Connecting 2 Thunderbolt displays using daisy chain with the Thunderbolt 3 (USB-C) to Thunderbolt 2 didn't work for me until I upgraded to the latest Sierra build. (10.12.13) now everything is working as it should. If upgrading didn't work for you I suggest that the adapter is faulty.