Someone cannot just "claim" an animal is a service animal. They don't have to prove it, but they're not going to randomly pull that out of their hat. And the overwhelming majority of SAs are dogs. I've never seen a service cat, but I suppose it could exist. Also, no airline can supercede federal law, so regardless of SWA's policy, they cannot require the handler to prove something.
That said, most people with SDs (myself included), do have some outward signs of the animal's status. In my dog's case, it is his vest with patches; his ID card showing his picture/breed/DOB/registry number/handler, and on the back has pertinent information; the handler card; an interstate health certificate every time we travel; as well as physician documentation declaring the medical need and supporting the use of the animal to mitigate that need. How much of this I choose to present is up to me, none of it is required by law. Almost all could be obtained by someone who wants to claim the animal is a service dog when it is not, but it's a lot of hassle and expense to go through just to hop a flight. And I don't think it would ever work with a cat, or that someone would try. Plus, there is a lot of training put into a dog to have them public-access ready. Most run of the mill animals will do fine in several situations, but a SD is far and away better behaved and trained than most dogs. It has to be. It also comes in handy when all the neighborhood strays hang out in our yard and I only need to tell the dog to leave it, and he ignores it completely.
I would think it would be just as, or more, likely that someone claims an allergy just so they don't have to sit near a pet, than for someone to claim their animal is a SA just to get on the plane.