Commissioner Gordon, Idiot or Liar?
There are only two choices possible regarding Commissioner Gordon not knowing Batman's real identity. Idiot or liar.
He clearly stated that he often ate lunch at Wayne Manor, and he was also seen in many party scenes at the estate.
Fact 1. - Alfred the butler was in attendance at most if not all of these events.
Fact 2. - Alfred the butler often answered calls from Commissioner Gordon on the Bat Phone, perhaps even more often than the caped crusaders, as they were often engaged in activities outside the room containing the phone.
Conclusion - Commissioner Gordon had to have heard Alfred's distinctive voice both in person and on the phone many, many time. Yet, he is never shown having the knowledge that Bruce Wayne is secretly Batman. In the TV series, he even states at one point that he wished to never know Batman's real identity so he cannot be held responsible for the information.
And no, later books, movies, TV shows that change this relationship do not count - we're talking 1960s TV show ONLY.
Idiot, Liar, or a really bad phone line? Pick one.