Yeah, real life skinning is fairly basic between animal types. Harvesting from insectoids would be pretty different I imagine, but having different skills for Skinning is pretty unfair to those who choose to go into that skill compared to, for example, mining. This is one area where having it as one skill might impact suspension of disbelief a little, but makes it a much simpler, more fun experience for those who choose to do it. Unless you plan on breaking down Fishing into different skills based on if you're fishing in water vs. acid.
That being said, requiring different tools for different animals sounds like a capital idea. That skinning knife may work great on rabbits, snakes, and frogs, but you'll need something a little longer and stronger to skin a bison or dinosaur, and you'll need something else entirely for properly harvesting a giant beetle carapace!