One does get the idea that the Mouth of Sauron is of great age, prolonged by sorcerous Black Numenorean arts. As for remembering his name, just look at Gollum, for the Ring had driven him mad and had (nearly) cost him his identity as well. One such as the Mouth, who willingly submitted to all that Sauron would teach him, might after an age or more, forget much, including his name.
Was the Mouth a Ringwearer? Perhaps it was not a Ring of Power, forged by the Elves. Did not Saruman also sport a ring on his finger? Might not another ring, the merest trinket, be made for a servant like the Mouth? Such a ring would not give power, but only physical life, at the cost of one's soul and the damnation of one's spirit.
As for the Mouth being allowed to use the name of Sauron, perhaps "Sauron" was not the name used. "Morgoth" was a name given Melkor by the Elves. "Sauron" must have been a like naming.
Sauron might well permit even this name to be used somewhat, particularly among his upper caste minions, such as the Mouth or Saruman. Perhaps it was his true name that the Mouth was permitted privately to use. Gandalf did not mind sharing his true name of "Olorin" among friends but Sauron would be more careful. Recall that even Sauron was not created evil in the beginning, and to some extent it may have pleased him to hear his true name on someone's lips from time to time. Saruman would have been one to know that name. Would Sauron have revealed it to the Mouth?
That the Mouth would use the name "Sauron" before the foes of Sauron, or any others, may not be remarkable in that the idea would be to instill or reinforce fear.
But there is another name of Sauron that may answer the riddle.
Recall also that Sauron for an age called himself "Anatar" the Lord of Gifts. This might well have been the name that a Black Numenorean sorcerer would have called him, who had worshipped Sauron as a god, perhaps even in that day. The entire point of this cult was the prolongation of life, the rejection of death (the "Gift of Eru"), rebellion against Eru's servants, the Vala, and the worship both of Anatar/Sauron himself in the flesh and of Melkor in the spirit. Sauron would certainly not permit those he ruled by fear to call him "Lord of Gifts."
But he might have let the Mouth get away with it. This may have pleased Sauron's vanity to be reminded of a scheme that had nearly succeeded, and that might have yet finally succeeded in that time, and so take away the scalding sting of the defeats that had left his hand first blackened and then bereft of a certain finger.
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