The originals were made around 1500. The new tapestries are being hung at Stirling Castle in Scotland. James V is known to have had unicorn tapestries there (of some sort, not necessarily the ones at the Met) during his reign in the first half of the 1500s.
The artists worked according to all that is known of the weaving methods and materials of the time.
The video is a must-see! (Follow link at the bottom of the main page.)
(I continue to be fascinated by tapestries ever since my experience a while back with helping conserve one.)