Assuming that "they" knew who it was in the photo, it does not follow that the person in the photo had anything to do with Oswald or the assassination.
Hoover was wrong that the person used Oswald's name; he either got bad information or misunderstood it (as he said in the call: it's "confusing" and it "appears": he wasn't sure). How would Hoover know what that person in the photo told the Soviets that he met? What his name was? Hoover told LBJ in that early period lots of wrong things, e.g., that Oswald was arrested in a shootout at the Texas Theater and that a police officer was killed. It was the type of misinformation that was all around during those early stages.
Is one theory that this person - who clearly isn't Oswald - impersonated Oswald AND THE SOVIETS never revealed it? They kept it quiet? They fell for it? But that's wrong: the Soviets themselves said it was Oswald who went there and that the person in the photo didn't identify himself as Oswald but was another person.
Or is the theory that Hoover and others manufactured this visit to Mexico City and then Hoover ON TAPE exposes the impersonation? And the tape is released?
So the theories are (1) the Soviets knew about this impersonation and kept it quiet and that (2) Hoover knew about the impersonation and reveals it on a phone call that is released? Is this what conspiracists are reduced to arguing?
Again, the three KGB agents/Soviet Embassy officials who met Oswald were shown the photo that Hoover mentioned. They all said the man never said he was Oswald. In fact, Oleg Nechiporenko said the man was a US Air Force sergeant who visited the Embassy and offered secrets to them. It wasn't Oswald.
Shorter: Hoover told LBJ all sorts of things that were wrong. This was one of them.
Here they are at the 1:10 mark interviewed in the PBS show "Who was Lee Harvey Oswald?" Once again, they all said the man they met was the real Oswald and not the man in the photo who supposedly said he was Oswald.