For my generation the TV dog was "Comet" from "Full House." We all felt a connection with Comet. When they relaunched the series with Fuller House, they even brought back dogs that were supposedly several generations descended from Comet. TV fans just want someone or something to relate to. It might be a person, a puppet or an animal.
Moose got more fan mail than any of the actors, which is a pretty clear indication that he was well liked. People love animals, which explains the famous animals like Grumpy cat. They can become obsessed with the animal actors just like they do the human actors in tv and movies.
Yes, television fans develop an affection for animal actors as well as humans because it is impossible not to. This is why we have pets. We treat animals like family, and animal characters are no different. They have personalities are important parts on TV shows. Moose was a great addition to "Frasier."
I don't think we do develop affection for animal actors. We grow in affection for the character they play, but not the actual animal playing the character. This is because we don't really see a difference in an animal if their of the same type (ie. boston terrier). But if the character in the show dies, we do feel a loss.
We are humans. Our brains are designed to recognize other individuals of our species. We are designed to relate to other humans. This is why we can recognize people we know or have seen before on television. When we see an animal, however, although we may be able to recognize its species, or even its breed, it is unlikely that we would be able to distinguish it from another without constant reminder. If an average human saw the dead body of the dog Moose, they would just see it as another dog, unless they had been told it was Moose. This is simply because our brains are designed to empathize with and recognize other members of our own species, not dogs, or any other type of animal. In order to feel a distinct emotional connection to it, as opposed to any other dead individual of the same species, we would either have to live for years with it, or be informed from other sources why we should feel this connection. Additionally, since animals cannot communicate with us in the same way that human actors do, we cannot really get to know them as individuals. We can pretend to know them, but really all we know is what we were told, or the personas they were assigned. Thus, it is impossible for television fanatics to develop a real emotional connection with animal actors that they see on movies, as the animals cannot properly express themselves to us, and we cannot truly recognize them as individuals or relate to them.