Sound recordings can be, and new ones normally are, protected by copyright. The copyright would usually be held by the person who made the recording, or that person's employer, not by the speaker if that is a different person. Use of such a recording without permission might well be copyright infringement.
But more clearly and directly, broadcasting a statement:
I am {performer} and you are listening to {song name} on my favorite station {station name }
without authorization from the performer would in many jurisdictions violate the performer's right of publicity, giving the performer grounds to sue the station. It might well also be false advertising, implying that the performer had endorsed the station when s/he has not done so. That would depend on the specific laws of the jurisdiction where the broadcast was made.
A suit over publicity rights or an action for false advertising would probably be simpler than a copyright suit in such a case.
I think that most if not all such announcements are made with the consent of the performers involved, and are probably recorded directly by such performers. (For one thing it is usually in the performer's interest to cooperate with stations and networks that play the performer's work.)
If such announcements were somehow artificially synthesized, but with the permission of the performer, and of any owner of copyright in any recording used, I don't think there would be any legal problem.
If an AI was trained to create a good imitation of a person's voice with9ut diretly copying a recording, I am not at all sure if there would be any copyright infringemetn under current law. That may be an area where the law will need to change to respond to the technology. But if such an imitation were used without permission to make the kind of statements discussed above, the personality rights issue and false advertising issue would still be there. Those do not in any way depend on whether the announcement uses a copy of a recording or not, those are both about the use of someone's name and reputation without authorization. In fact, even if the announcement did not pretend to use the performer's voife, those woudl still bne an issue. Suppose the announcer saids, in his orm her own voice:
{Performer} said to tell you that s/he is glad thit his/her song is being playing on his favorite outlet {station name}.
There is no technical fakery there, deep or shallow, but if done without authorization it is still a problem, or would be in some jurisdictions at least.
If technology is used to create a plausible imitation of someone's voice, but it was not distributed with any claim, direct or implied, to be that person, then the case is different. I suspect that in most jurisdictions there would be no grounds for legal action, just as celebrity imitators do not need permission as long as they don't fake endorsements.
