We often consider a social organization by focusing just on the one dominant species, whether we're looking at bees or humans, but of course our human society is made of more than humans. We could choose a perspective that certain animals are rightful and integral parts of society, though if we really focus on interdependence our perspective will swell to cover a whole ecosystem, and that would be no focus at all because it may not be useful to consider a tree a member of our “society.” So where should we draw the line?
I think it interesting to consider a few choice animals as actual members of our society. Although certain animals are noted for performing useful functions — functions that can be replaced by machinery and modern technology — our connections with them really go much deeper than that. Let's just consider dogs and cats.
Dogs perform some useful functions, but they can only do so by intense socialization with their owners. Dogs can assist in hunting, which is useful, but it makes no sense to think of a hunting dog that hasn't been trained while spending a lot of time interacting with humans. Dogs are commonly kept for security, but they can function so only when they have bonds with the members of the households they live in. For purposes of alerting their owners about activity near the house, they can distinguish between family members and strangers even before they ring the doorbell. Many dogs also stand ready to intervene in case a stranger attacks their owner.
In addition to simply being useful, dogs are known, of course, to be “man's best friend.” People enjoy the companionship of dogs, and they care for them and about them almost as if they were human children. Having your dog with you in public facilitates socializing with other people because it attracts attention and advertises a common interest to other dog owners. In fact, sometimes neighbors talk to others' dogs and not to each other.
Cats are different because dogs and humans have socializing behaviors that are more similar, but cats are still useful and enjoyable companions. Cats are an original form of pest control, killing all rodents and insects that come in their way. Cats made possible our civilization as we know it since they protected our stores of grain from rodents in the early days of the agricultural mode of civilization. That function isn't so important anymore, but we still seek the companionship of cats, and cats still seek human affection.
Our pets do not simply perform work. We have personal relationships with them, and we care for them as members of our household and not simply as items of personal property. They are the non-human members of human society.