To elaborate a bit more on what Turnbull has said, when watching that scene, I always felt that when Michael says that to Vincent, he is telling him that he, Vincent, will now be accepted by him as a Corleone! Remember Michael tells Vincent that if he chooses this life, he must give up his daughter, and Vincent agrees. I believe that it is right after this exchange of dialogue, that Michael tells him that he can now call himself Vincent "Corleone."
Then when Michael slips out of the room and Vincent is sitting in the chair, we see Neri and the two other guys bend down, similar to what was done to Michael in GFI, kiss his hand, and if memory serves me correctly they refer to him as Don Vincenzo.


Don Cardi