Is there a difference between being allergic to cat hair and being allergic to dog hair? Or is it just pet dander that people are allergic to? Is there a difference in cat dander and dog dander?

Yes, there's a huge difference between cat and dog dander.

A person that's allergic to cats, isn't necessarily allergic to dogs. Just like a person that's allergic to peanuts, isn't necessarily allergic to all nuts.

Cats are cats, and dogs are dogs. Just like people are people, and elephants are elephants. Big difference between the two.

Yes, large difference, some people are allergic to both, many are not