I don't know whether there is a grammatical rule.
But there's probably a rule of etiquette.
Yesterday, I spent $166 on medicine for our Jack Russel and then took her to the dog park for 3 hours and then gave her a bath and then hand dried her for 15 minutes. My wife brushes her (the dog's) teeth twice a week. Our lives are seriously affect by this silly creature. We do our best not to get delayed getting home, because we know she will hold her pee until she is wimpering and then dribble all the way to the grass.
People love their dogs as if they were humans. It's jarring to them to hear someone refer to them as "its," as if they were just things or possessions. I'm not one of those kooky people who refers to my rough beast as a "fur-kid," but still if someone in the dog park referred to one of them as an "it," I might suspect he was a bit touched.
I haven't read any of Herriot's books, but I've seen them - and I assume he's a vet. Vets, if they want to keep business, will be in the habit of expressing compassion - for the animals and for their keepers.
