Have not seen the show, but I've had a few people ask me about getting their pets stuffed after they die - I just tell them we don't do that, if they want to seek out a taxidermist they are more than welcome too. I think it's weird, but I'm not going to say anything - it's their business, not mine.
We had a prof in vet school who was a big game hunter - he had his trophy room set up in his house all full of taxidermied animals, heads on the wall everywhere, etc. We called it "The Dead Zoo".
We once had a client call up because their dog was very old, doing very poorly, and they figured it was time to have her put to sleep. They were asking about prices and we told them the fees, and then costs of burial vs. cremation, etc. They said they would take her body home - which is fine, because lots of people do to bury them at home (which may not be legal, but I am not going to report anyone). But then the lady said that it was so they could SKIN her! I was like, WHAT??? She said, oh yeah, we've done it with all our other pets! And I am picturing their house, and pelts all over the place, of tabby cats and maybe a Dalmation, or whatever. I hope she was joking!
I was joking with DH the other day about getting my older pug stuffed when he passes and making him look like he's just sleeping in his bed, since that's how he is 90% of the time anyway. But I would never do it.
One time I had a guy who had to have his dog put to sleep but was quite distraught about it. He kept saying he could not imagine not seeing his dog's beautiful eyes looking at him every day. Then he asked if we could save the dog's eyes for him (the body was to be cremated). I was like, uh, no, sorry (while I am thinking, you want to see them all bloody and floating in a jar of formalin?).
Another time a guy had to get his older dog neutered because of the dog having prostate issues and only neutering would resolve them completely. He was reluctant, but agreed. Then he asked us to save the testicles for him so when the dog did eventually pass on and was buried at home, he could be buried with them as well (we did honor that one - he saved them in a Ziploc freezer bag and took them home when he picked up the dog).
So if you think that is freaky, work a vet hospital for a while, and you will really see how freaky people can get. Just when I think I've heard everything, someone else really throws me for a loop.