While I do not believe in prying into the private affairs of my fellow Hogwarts staff members, I feel compelled by your question to offer a statement regarding companionship.
My dearest friends, who no doubt you've heard of, and I had many encounters with the caretaker in question throughout our Hogwarts years. We were students and, on many an occasion, behaved as animals would with complete disregard for the state of order that men like Filch work so diligently to uphold.
My friends and I had each other. We were our own world. Not much else mattered. It is my belief that everybody needs somebody; I shudder still to think how my life would be had I not come to find Hogwarts and, with it, my friends.
While I could not go so far as to suggest that the frequent bizarre sightings of a large, black shaggy dog causing chaos around the castle, or muddy dog prints suspiciously close - and leading up to - Filch's meager office (as if a spectral hound meant to torture the poor man with the promise of never-ending labor) during our school years would have solely guided Filch toward the feline (as opposed to canine) persuasion, his choice in species for pet is quite obvious...
Cats are tidy beasts. They pick up after themselves. Dogs, stags, rats and wolves, I fear, do not.