I seem to be a rare exception to this (at least, I get the impression it's rare not to feel attached to the dog, hence of course why films use animals to tug on the emotions of the audience). Attempts to evoke concern for them fall somewhat flat for me. If a film tries to make me care about a dog or a horse, say, I simply can't; unless it's integral to the story. If the film were
about a person's relationship with a dog or a horse, that's different. I'm a rather sentimental person, after all, and I can easily accept other people's depth of feeling for animals. But when a film includes a scene like the one in
Independence Day sojourner mentions, I'm a little confused and perhaps a bit resentful when it's implied I should value a dog as much as (even more than?) other humans.
To be honest, though, something similar actually applies to humans themselves in disaster movies, etc. To invest in a character or a small group of characters is easy if they have compelling personal problems to work through, but when their main problem is "the asteroid", how are they particularly worthy of my attention? More than anyone else, I mean.
Everyone has that problem.

I can never wrap my brain around why I'm supposed to care for the protagonists specifically. What about the three hundred people we just saw killed by the tsunami? The protagonist is no less worthy, of course, but no more so either. Throw in a dog and it becomes a strange exercise in reflection on human group dynamics and my own biases.
On the other hand, in terms of books like the one
Miss Chicken is reading, I wonder if it might have something to do with a non-sapient animal being a blank slate onto which our general sense of concern, stemming from the situation the characters are in, can be projected. After all, the human characters, likeable or not, are subject to our understanding that relating to other people is a complex and somewhat contradictory exercise. There's always some degree of baggage affecting how we relate to someone, and perhaps the "purest" appreciation of the emotions the book evokes results when we can disregard that basic empathy and just treat the character as an object - an object of concern.
I certainly remember a book I once read about a group of people on a quest or journey, and the one "character" I hoped would survive to the end was their moody old donkey. Though maybe that's because I wasn't too invested in the characters, but the plot was good.