It is about a dog. I posted it here some hours ago but I could not find it. Paul himself told it is for his dog Martha, Here are his words: "You silly girl, look what you’ve done,’ all that sort
of stuff. These songs grow. Whereas it would appear to anybody else to be a
song to a girl called Martha, it’s actually a dog,…

It is about the dog. Paul has affimed that after reading it was not. He said he wrote the song indeed for his beloved sheepdog. People who said it was not, are people who never had dogs, for sure. Paul loves animals in general. If you look for it you will find his interview saying it was for Martha and nobody else.