I agree with DC and Turnbull. I believe that the scene you refer to is Michael's way of accepting Vincent into the family, not necessarily The Family (although I do believe that the two go hand-in-hand in this scene). At the beginning of the movie, Vincent had to crash the party. Only Connie accepts him as a member of the Corleone's. Joey Zaza calls him "bastardo". He was definitely an outsider. In this scene, Michael was accepting him as a Corleone and recognizing his blood ties. This recognition gave him the ability to be accepted as the heir apparent by the others.

While we see how Michael's life ends in Sicily, I've always wondered how Vincent ended up. Did he take over after Mary's death?