I was thinking a couple of cases where people regularly brought their dogs into the church.
One was a homeless man with a fairly large mutt. They were inseparable. He would sleep outside if he couldn't find a room to rent that would allow the dog. He shared his meals with it. And he brought it to church. He sat alone in a back pew with the gate closed. Most people in the church were unaware of the dog's presence except when it accompanied him up the aisle when he went forward for communion. When this man had to go into hospital, his big concern was that his friends look after the dog. When he died, his friends had a heartbroken dog put down.
And the other was a middle-aged woman was obvious mental and physical challenges. She brought her small dog with her and held it on her lap during worship. I understand that she lived out of town and got driving on Sundays to attend church and spend the rest of the day with her elderly parents. Having the dog with her gave her a sense of security and relieved her of the worry of leaving it alone for the day. Again only those who sat near her were aware of his presence.
My third example of a companion animal is more recent. One of our church members moved into a garden apartment in an assisted living facility. He was surprised when told that he couldn't bring his small dog with him. He went to his family doctor and got a paper stating that he required the dog to calm his nerves. So he now has a companion dog: so far he has not brought him to church and I don't think he means to.
Dogs {and cats) can provide valuable services calming the nervous or excited, providing companionship for the lonely, giving meaning and purpose to people's lives.