I agree, only hunting counts as a skill.
Having said that, the dog hunting thing is stupidly lucrative.
I think the difference is that Cats can learn hunting from the toy in the toy crate but dogs have to be taught by a sim. I haven't tested it, but my partner had a kitten 'watch' a rodent/bird and when the kitten aged up, they had a high hunting skill.
My dog will hopefully have puppies soon so I'll do some testing and let you know.
A