When I went through this procedure I asked my doctor the same question – the answer is that theoretically you could do the same thing on your own, but immunotherapy is a very precise dosage that changes over time that would be very difficult for you to match. Plus there’s the safety issue of getting a reaction (which did happen to me).
As a person who “was” allergic to cats, and has had over 400 cats in the last 20 years, the best way to get over a cat allergy that isnt severe(as you described in aonther comment as runny nose and watery eyes) is to just bring home a cat. After 2-3 months of living with the cat, your body will adapt. The biggest issue with cats though is cat litter. If you live in an apartment, biodegradeable flushable cat litter might seem like a good option, but it will tear your sinuses apart. Get the chemical cat litter. The same thing goes for dogs if youre allergic to dogs too. Get a dog, after 2-3 months of insane sneezing, youll stop being allergic. Its how ive got over my allergies.