That has to do with the general assumption the dog is innocent. This, in turn, has a lot to do with how the dog looks. With such animals, you can make them look cute and cuddly. This can trigger emotional responses normally associated with human children. Small dogs too, but especially kittens are prone to this. Cats are very anthropomorphic, meaning they can (and will, as anyone who ever owned a cat will tell you) make you think about them like of a child, and thus provoke a similar emotional reaction. But it only applies to "cute" animals. A random stray mutt won't do. A small, dirty, meek stray dog will cause a much stronger response than a large-ish, dominating one. Indeed, certain animals can get by on "rule of cute", but it usually requires designing the animal in a certain way.
You can do that with humans, too, BTW. The general rule is the younger and "more innocent" someone is looking (blond hair, immaculate face, frail body), the worse the violence against such a person comes out. Gender does matter, but you could depict a boy like that and he'll get a lot of sympathy, too (that's all assuming no other development, of course).
With humans, it's widely accepted that a random adult is both not as vulnerable as an animal and not as innocent as well. It's a general rule. Attacking an easily recognizable religious person, for instance (for example, a priest or a nun) will, by default, get a stronger reaction because of this "innocence" association. A doctor or nurse likewise, or someone sick. We need a reason to assume a random bystander is "innocent" or sympathetic, but there's a number of easily recognizable groups that we automatically make this assumption about as well.
The problems with using any of the above is that it's easily to fall into a cliche. In particular, we all know saintly, innocent, blonde boys/girls from fairy tales. Likewise, while it's possible to have a cute dog work in such situation, it's possible to overdo it. A kitten is likely to plunge the scene straight into parody territory, Pratchett did pull it off in a really touching way, once, but it's hard. With a human, it is, I think, much easier to have it come off seriously, yet touchingly. A human child, particularly, enacts a very, very strong reaction (in fact, too strong for most game designers to use), but a young adult can work, too.
Also, there's a matter of how the violence is depicted. The above assumed "casual violence", as in, walking around and kicking dogs without even thinking about it. If you want to show someone reveling in violence, animals are straight out. This just comes off as silly in most cases, though certain genres can get away with someone exploiting animals (like horses) to death and having a hint of sadistic glee about it. When you show someone who hurts someone and enjoys it, then you pretty much have to have a human.
The default emotional reaction is pretty different regardless of how much buildup the dog/person has.
If there's buildup and the audience still has a "default reaction", then it's a sign it fell flat. That's writer's fault to ensure that we care about the characters. If there's buildup, if the character is rounded out enough, then the reaction to that character being hurt will (hopefully) be what the writer intended, not anything default.