Dogs and cats, as predators, are both brighter and, at least in my mind, more adjusted to dealing with injury. A horse(and other similar herbivores) with a broken leg was predator food pretty much no matter what. Meanwhile a wolf at least has a chance, the rest of the pack can support it, at least for a time. As long as they have food, water, and a place to go to the bathroom, they're fairly willing to rest and let wounds heal. They've also been domesticated to the point of trusting humans more.
As for humans, I think that it's a combination of factors. We're the guardians, caretakers of animals, who can't make the decision themselves. Since our medical technology is still limited, we must make a decision that includes possible suffering. Because everything eventually ends up being economics as well, we must acknowledge that we can't afford heroic life saving efforts for every animal, especially when you realize that you're only extending the inevitable, and in consequence, their pain.
Humans are different. At least in theory we're all our own master. We can make our own decisions. Children are precious to us, and will eventually be able to make their own decisions, so we attempt to save them as much as possible, including heroic efforts. Even for adults, there's always the hope that we'll recover. Euthanasia does happen however, under the cover, in secret. High doses of opiates dull all pain, but also shorten lifespan. As GoRon says, this is fairly common, but untalked about, much like fatal 'gun cleaning accidents'. There have been some issues brought up, mostly in the EU, about this being done by the doctors themselves, however.
I believe that the second facet is our religious heritage that prohibits and demonizes suicide. Thus, we have humans, able to make their own decisions, and a general cultural policy that prohibits the taking of one's own life. Because of this heritage, a doctor makes the assumption that anybody would choose 'all life-saving measures' unless there is ample evidence otherwise, as the doctor is acting in the 'patients best interest' if he or she is unable to make the decisions themselves. That and the ancient oath 'commit no harm'. However, our technology has reached the point that there are relativly few people whose usefull life can't be extended through injury or illness. Even those completely invalid. Even those who have lost their mental facilities. Thus the importance of 'living wills' and medical powers of attorney.
