Why is is OK to stuff "game" trophy animals, but not a pet after it died?

This is a sequel to this question I just posted.

I think stuffing any animal that was once alive is in very poor taste, period, but stuffing a pet takes it to a different level in my heart and in my head…but I can’t quite put my finger on “why.”

Is is simply because we knew the animal personally, or the person who had it done knew the animal personally? Why would that make it any different than stuffing an animal that someone didn’t know “personally”?