You need to listen to the entire line of dialogue or you won't understand the meaning. What Gandalf says is:
Indeed I am Saruman, one might almost say, Saruman as he should have been.
He clearly is not claiming to be the same person, or even to have taken Saruman's name. He's claiming that he is the wizard that Saruman was supposed to be when they were sent to Middle Earth.
The five wizards were sent to Middle Earth for a reason, and given "orders". Saruman was chosen to be the leader of that group, and to make sure that the others did not stray. He was also expected to be the one to help the people of Middle Earth fight Sauron. Obviously, he didn't do any of those things.
So, after Gandalf died fighting the Balrog, he was sent back to Middle Earth to take Saruman's place, and to do Saruman's job for him. In essence, Saruman had been "fired". There is a strong implication that Saruman was stronger, or had additional power, that Gandalf the Grey did not have. But now, as Gandalf the White, he was Saruman's equal, or even the greater of the two.
In other words, Saruman had a role to play, and he failed to live up to that role. Now, Galdalf has returned to Middle Earth to be that person instead. He was trying to explain to his companions what Saruman -- who used to be a close friend to Gandalf -- was "supposed to be", before he was corrupt, and assure them that he (Gandalf) was going to make up for Saruman's failings.