In the Godfather III, when Vincent is made Don of the Corleone Family, Michael tells him that he will now be known as Don Corleone.
But, if I am not mistaken, weren't Dons whose names were different then their Family names just called by their real names.
For example, after Carlo Gambino dies, and Castellano was made Don, he was called Don Castellano, not Don Gambino right?
Is this another Puzo thing, or was this a legitimate thing to do?

Well, normally when a son or daughter is born, they inherit the father's name. But since Sonny died, I'm assuming Lucy just saw fit to give Vincent her last name. But by tradition, Vincent's name should have been Vincent Corleone. Maybe that was part of Mike's thinking. Also, I don't think Mike would want to defile the family anymore by letting Vincent keep the name of a woman that his brother had an affair with.

Vincent would have been "Don Manchni" wouldn't sound right

Michael said, "Nephew, call yourself Vincent Corleone." He didn't say, "call yourself Don Corleone," probably because Michael still regarded himself as the legitimate Don Corleone. Had Michael died and Vincent taken over, he probably would have called himself Don Corleone.
Carlo Gambino was known as "Don Carlo" by all. Castellano was known as "Paul," or "Paulie" by those reporting directly to him. I don't know what the lower-level people called him.

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

Ok, here is the dialogue for those scenes :

"Nephew, from this moment on, call yourself Vincent Corleone."

<Vincent kisses Michael's hand. Michael motions for him to sit down as Calo, Al Neri and Armand acknowledge their new Don.>

CALO (kissing Vincent's hand)

"Vincenzo."

VINCENT

"Grazie, Calo."

AL NERI (kissing Vincent's hand)

"Don Corleone."

ARMAND (kissing Vincent's hand)

"Don Vincenzo Corleone."

<Michael, watching, turns away and leaves with Connie, the door closes.>


Don Cardi

Oh, well I thought that the other guys called him Don Corleone. But you could be right, Don Cardi.

And about the whole Paulie thing. The called him Paulie? Well, I would thing that that was a little disrespectful-I don't know maybe I've been watching to much of The Godfather lately.

The quote that I posted is directly from GFIII script and upon reading it I now remember the scene very well. As for Big Paul Castellano, I believe that it was by HIS choice that he NOT be called Don Castellano. He was from a different mold than the old timers. Big Paul was more of the business mind, and wanted his family to be run like a legitimate business. He wanted to be known as a businessman! He and his bodyguard did not even carry guns. It was this very attitude that resulted in his downfall as the people under him did not agree to his theory that they should run things like a legitimate business. He also was a hypocrite as even though he felt that way, he still took his "share" of what the people under him earned, no matter how they earned it. But he did NOT like any one of his people portraying an image of a gangster/street guy over the image of a legitimate businessman. To make matters worse was that by Big Paul having the "business"man's attitude, many other families started to move in on the action of Big Paul's people that were under him, and instead of fighting them the Cosa Nostra way, he sat down and negotiated with them, without consulting the people under him who were being affected, and made "business" deals allowing the outsiders to take a piece of the action, cutting in on the profits of those under him and at the same time still taking the exact share that he took before! So in a nutshell the guy wanted to run things like a businessman and did NOT want to be known or reffered to as Don Castellano!

Don Cardi

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?

Why shouldn't he? It was his main goal in life, and I always believed that business interested him much more than stupid love affairs.
Maybe Michael might adopt him and give him his name.

Re. Vincent "taking over" the family:
One of the subthemes of GFIII was how Vincent and Michael used each other. Vincent cleverly buttered up Connie to get close to Michael. When Michael seemingly took Zasa's side and ordered Vincent to "make his peace," Vincent forced the issue by biting Zasa's ear, thus making Michael choose between family or "ally." But though Michael appeared to condemn Vincent's hot temper and violence, he was using Vincent as a kind of blunt instrument to let others know that he could still summon drastic action, if needed. Then he put Vincent in mortal danger through the ploy with Altobello. I believe that, if Michael had had his way, Vincent would have been little more than an arms-length underboss--a kind of head of security and an enforcer, to be kept increasingly distant as Michael became increasingly "legitimate."
As for Castellano: Though he wanted to be called "Paul," or "Big Paulie," he really didn't have the respect of his men. He violated Mafia "rules" by carrying on with Gloria Olarte, his Columbian maid, under his own roof, with his wife and daughter resident. He squeezed his people unmercifully, and involved himself in petty matters that should have been handled by capos or lower. He seldom left his Staten Island mansion to meet with his people, demanding that they come to him (and only a few were permitted this "honor"). He also disrespected his esteemed underboss, Neil Dellacroce, by failing to come to his wake or funeral. The last straws: after Dellacroce died, he dissolved his crews (including Gotti's) and named his driver, Tommy Bilotti, a stammering, bat-wielding thug, as his heir-apparent. Castellano's RICO indictment caught him making denigrating remarks about the other NY Mafia Dons. They feared that he'd turn rat rather than spend the rest of his life in jail without his beloved Gloria. That's why they approved Gotti's murder of Castellano.