Personally I see two elements in this picture. One is the human and the other is the animal. I don't understand why a Christian can berate another person for what they believe. If a person loves Christ and loves like Christ, and seeks Gods ways in their life, what they think of an animal today will be consistent with what our Lord wants them to think and act like. You can be kind to a person and keep your own beliefs to yourself. However, to be realistic (where it might not be wanted,) we don't read a lot about pets in the Bible because there were no pets as we know them today in the times of the laying down of the Bible. Unless people lived in luxury (i.e., the Pharoh, Kings, Royalty) animals were in proximity to man for food or labor for farming, or for feeding the daily life needs of persons. (Soap, glue, sinew for tread, leather, clothing, etc.,. Not a piece of the animal was wasted.)
Or they were prized and owned property such as the Kings stags. Or, they were pests or varmints or dangerous predators. If the dog was feral he was a nuisance animal. If it lived with a family it was trained for hunting. A feral cat was for keeping the grain lots free of mice. Animals were permitted to live close to man because they had a purpose. Still, one could not afford to treat an animal like a person. It was another mouth to feed and body to tend to and if it didn't return more than it gave it was of no value to the humans struggling from day to day to live. I am sure there were kids who wanted to cuddle a rabbit or puppy or kitten, but animals were creatures. And, we have living examples of how animals might have been treated those many years ago because they still live the same way in the remote villages of Africa and similar places where animals are treated pretty much the way they were thousands of years ago.
Now that we established at least a small parcel of why pets are not mentioned in the Bible I will add that praying for the comfort and blessings of the person who takes care of the animal is not wrong. In fact, if that person is in anguish over an animal or a friend, over the loss of those thousands in the World Trade Centers whom they probably never knew, of the surviving children made orphans because of a drunk driver, anguish of the human soul is anguish and praying for them is all about the Love Christ teaches us. Even for the mother who's anguish comes to the surface later, yes, we must pray for her too. Regardless of the reason for a person hurting we can pray for that person to find peace and yet, we can still hold onto our own beliefs about such things and still be true to the Lord.