Good question...I hope you aren't asking because your own pet is ill!
Anyhow, the answer to your question is: No, cats (and dogs) do not catch "the Mumps." Mumps is a virus that only affects humans. Like most diseases that affect people, cats and dogs cannot catch them. If you have an animal showing mump-like symptoms, you really should take it to the vet because it has got to be a sign of some other illness. It could be an abscess, an infected bug bite, or a tumor. At any rate, it should be diagnosed by someone qualified.
On sort of a sick and horrible note, there have been some experiments in laboratories with cats and mumps viruses. I have posted a link if you want to read anything about that.