*Note: dogs are a very different case. We REQUIRE dogs to meet any dogs already in the home.
In short: you won’t gather any information and you will just stress your cat out.
Cats take time to get used to new environments and new animals. Just coming to the shelter would be overwhelming for your cat. There is always so much going on, dogs barking, and a million different scents accosting their super-sensitive noses. A short meeting at the shelter will very likely not help you determine if the animals will get along. Cats are also very susceptible to upper respiratory infections, especially when they are stressed. Colds are very common in shelters, thus you would be exposing your cat to that as well.
Meeting another cat they likely won’t get along right away. The vast majority of cats take time to get used to a new feline companion. There are rare cases where they are immediately friends, but that is extremely uncommon. This puts your cat and the shelter cat at risk of getting into a fight. Most cat introductions require SLOW introductions! We have a full blog post on how to do that here.
Meeting a dog is likely to go much the same. Some of our cats have lived with dogs before and will nuzzle right up to them, but even in those cases, there is a good chance they will be unsure of a new dog. Some we have no idea if they have ever been around dogs, so they might be terrified. This also puts your dog at risk of being hurt as well as the cat. Cat to dog introductions also should be SLOW! This blog post also discusses how best to introduce a cat to a dog.