Yes, there were popes that were married, and yes, there were popes that fathered children. But, I doubted whether any of those would have carried a surname ‘Pope’.
So, I had to go looking. Appears it was a ‘not too nice’ nickname that stuck! Yikes. Indicated a pompous person that acted like a pope.
Yes, a possibility, that it might relate to those with a church, in particular bishopric, connection. But, the consensus is that it was most likely a nickname to start with. How a nickname eventually becomes a family name is a whole another conundrum.