To your question: How do I know if I'm allergic to cats?
The fact that you are itching like crazy at work is a sign that you are allergic to cats. Now, you may want to talk to the cat lady about your uncomfortable situation. There is a product out there called 'Allergy Relief from Cast' and I swear by it. It is not expensive and it's just a matter of rubbing the cat with that product once a week. I guarantee your allergies will go away.
The allergy would be to cat dander, which as Linda said, is actually pollen and other irritants that migrate to cat fur. The symptoms are those typical of airborne allergies and rarely cause skin itching. However, if the cat has fleas that could certainly account for your itching.
If you work in a house where the lady owns cats, and now you are itching, you could be. It could also be an allergy to something else that you are exposed to in her house (for example, if doing cleaning, maybe you are allergic to the cleaning product used in that location but nowhere else, or if you were a painter and using a specific product in her house that you don't use elsewhere, it might be the product). Cats have saliva that gives some people a strong allergic reaction. They lick their fur, it coats the fur, the fur floats around and is on furniture, etc. and in the air, and some sensitive people sneeze just from being in the house where the cats have been. I've had it happen with friends, and I only have one cat. (These were friends who knew they were allergic to cats, so it was obvious what it was.) But the usual way for a person allergic to cats to present with that allergy is by runny eyes and sneezing, not itching. Itching is usually from skin contact. Have you changed your laundry detergent or soap lately?
Usually people who are allergic to cats are really allergic to the pollen that can collect on their fur. So it's more like a hay-fever kind of allergy than an allergy to the cat. If you are itching because you work with someone who has cats...maybe it's the perfume she is wearing. I noticed you joined the cat forum, so welcome.