If you know 1 dog, you very likely know most dogs.
If you hear of one dying, it's quite sad. It's very very easy to think of that dog and imagine your own.
With other people, it's a bit different.
People are so different from each other that you can't as easily imagine 1 dead person as being someone you know dying.
All deaths are tragic. Human and Animal.
But Animals are simpler than humans and easier to connect to. It makes sense that you'd be upset by the death of one much more than the death of a human stranger.
I personally pretty much never get upset or cry at the deaths of humans in films.
But if a dog dies in it, I'm blubbering like mad.
Plus, it's easier to not feel sad at the death of a bad person.
A criminal, or terrorist or whatnot dying isn't all that sad. You're more upset for the death of innocent civilians, say.
To some extent, it's definitely the same with animals.
Almost all dogs are just innocent creatures.
But if a dog that has attacked or mauled a child is put down, you don't really feel too upset about it.
Think about how many people in that "My neighbours dog killed all my kittens" topic a few weeks back had no remorse at the idea of killing, or even torturing a dog that had done a bad deed.