Allergy symptoms are caused by the immune system reacting to something that's harmless. People that don't have the allergy don't have an immune reaction. Pets don't get sick from their own dander because of course they wouldn't naturally have an immune system reaction to their own bodies (well, unless they had some autoimmune disease.)
The immunization is training the pet's immune system to have an unnatural reaction to the allergen that causes the immune reaction in allergic people. It's more surprising to me that doing this doesn't cause the animal to have some sort of symptoms, as having the immune system react to your own body is the root cause of autoimmune disease. I can only conclude the allergen is non-essential to the animal.