Talk about the importance of context! When I read the dialog in question, I laughed out loud. Then again, I've seen a lot of Monty Python: I could imagine their frantic mannerisms, and their madcap humor. In fact, their offbeat humor is so unique that it's even become a word in some dictionaries:
Pythonesque (adj.) denoting a kind of humour that is absurd and unpredictable; zany; surreal
If you want to learn English from watching Monty Python, then remember: any time you find yourself confused, you should probably be laughing. This comedy group loves to mix the inane with the intellectual, to mix slapstick with a unique brand of brainy humor. Only Monte Python could make the subtle difference between contradiction and argument incredibly funny.1
No, telling someone the wrong name of a city is not a pun, but that's what makes that part of the sketch amusing. The pet shop owner apparently has a hard time differentiating between a dead parrot and a sleeping parrot – and between a pun and a lie. Obviously, the parrot is dead; the idea of the wrong city name being a "pun" merely underscores how far the pet shop owner will twist his words rather than concede error. Whether he's a patent liar or completely out of touch with reality is for you to decide, but I recommend you pick the alternative that helps you laugh the loudest.
1You can watch the classic argument v. contradiction explanation unfold here, or read the script here.