So if a human hero dies, everybody panics, but if his or her equally popular animal hero dies, nobody panics?
Am I understand this logic correctly?
Not exactly. From a plot standpoint, animals tend to be "expendable" in that the main human cast remains fully functional if the animal dies, whereas if one of the humans dies the team loses an important skill set. So animals are "safe" targets for killings because upset the viewer/reader (in a good way), but don't introduce a handicap the writers have to work around.
Also, in some circumstances the death of an animal can be less traumatic to the audience than a human's death would be (especially if the method of death is particularly nasty). Think about how much of an emotional kick Serious Black's death inflicted, and was designed to, versus how much kick Hedwig's death inflicted, or Aragog's.
Of course, this doesn't apply in stories where the protagonist is an animal. The death of one dog in a sled team will be a bigger deal if the protagonist is one of the other dogs than if the protagonist is the musher.