A) Have you still not learned how to dress yourself? Fine. B) You don't have a dog and fetch the stick yourself. Fine, but I haven't heard this. What I often hear is You don't have a dog and bark yourself.
A is apparently referring to B's mother or servant, who dresses him so that he doesn't have to dress himself. So B has never learned to dress himself. A points this out, and B, unconcerned, says: If you have a dog, the dog fetches the stick not you, and if you have a servant, the servant dresses you, not you.
It is Britain, and yes, it is a conversation between a master and a servant. I haven't seen exactly this as an idiom, and I was thinking about whether this is an idiom or not. Thanks for your reply.