I'm sure that a lot of the animals us hunters have killed and subsequently handled had dangerous microbes or bacteria or germs or who knows what, either on the outside or the inside. I killed a coyote once that had rolled in human feces (he got trashed, not skinned.
I have made it a point to wear heavy duty (but throw away) rubber gloves for over forty years when gutting or skinning my prey.
Just use some common sense and be careful. In some ways touching a public bathroom door handle when exiting can be as bad or worse.