Not a lie--an "exclusive interview" is with CNN or MTP alone, as opposed to a press conference.
If the interview is with Cooper and the CNN reporter, even if Cooper is being interviewed elsewhere by someone else, both are still exclusives because no one else is in on that particular sit-down.
That is a new one on me an exclusive interview means to me that this person has not been interviewed on this subject by anyone else and that they are stating something that they have not stated elsewhere before.