The Animus subject played by Desmond Miles in Assassin's Creed is Subject 17.
Abstergo Industries assigns Desmond the designation Subject 17 because he is the seventeenth person used in its Animus research program. His genetic memories allow the machine to reconstruct experiences belonging to his ancestors, including Altaïr Ibn-La'Ahad and Ezio Auditore.
The designation is important because Desmond's sessions are not simply entertainment or ordinary historical recordings. Abstergo is searching genetic memories for information connected to the Pieces of Eden and the First Civilization. Desmond's unusual compatibility and ancestral lineage make him particularly valuable to the project.
Subject 16 is a separate Animus participant whose fate and fragmented messages become important to the wider story. Subject 12 and Subject 4 are not Desmond's designation. The number is therefore a precise piece of series lore rather than a reference to his position in the Assassin Brotherhood.