On the night of Nov 22nd 1963 Hoover was told by his FBI agents in Dallas that the voice on the Mexico city tape they had received did not match Oswalds voice.
But could it be that Oswalds voice just sounds weird on tape compared to hearing him speak on film or in person? Check out the following odd recordings of Oswalds voice taken in Minsk that to me sound nothing like Oswald (especially the first recording):
Hoover was given incorrect information, misinformation that was later corrected in followup messages to him. The FBI agents in Dallas testified that they heard no tapes of any phone calls (see the Lopez Report of the HSCA for details). That's because there were no tapes sent to Dallas only transcripts of two phone calls.
Here's an excerpt from the HSCA report:
"[Dallas Special Agent in Charge Gordon] Shanklin stated in a committee interview that no recording was ever received by FBI officials in Dallas. Moreover, former FBI Special Agents James Hosty, John W. Fain, Burnett Tom Carter, and Arnold J. Brown, each of whom had conversed with Oswald at one time, informed the committee they had never listened to a recording of Oswald's voice.
Finally, on the basis of an extensive file review and detailed testimony by present and former CIA officials and employees, the committee determined that CIA headquarters never received a recording of Oswald's voice. The committee concluded, therefore, that the information in the November 23, 1963, letterhead memorandum was mistaken and did not provide a basis for concluding that there had been an Oswald imposter."
The evidence is that, as was the usual practice of the CIA, that the tapes of the calls were erased and used for future calls. There's been a dispute about this ever since. Some people - including David Slawson who worked for the Warren Commission and who investigated the Mexico City angle - said that the tapes were not erased. Slawson, who is still alive, has said both that he heard a tape (the quality was very poor) and that he didn't listen to any and that he only read transcripts. He's been all over the map on this. His colleague, the late William Coleman went with Slawson. He said that he didn't remember hearing any tapes but that if Slawson said so then he must have (this account is in the Philip Shenon book on the Warren Commission).
In any case, the FBI agents in Dallas, e.g., James Hosty et al. all testified that no tapes were played for them. Were there tapes that the CIA kept? I think not but the evidence is mixed.
BTW, I took out the link to the video to save space.