It is highly unlikely that a dog would have allergies to its' own fur, but it could be having an allergic reaction some other fur. Or even something you use to clean it's bed or other household product.
VERY rare, but can happen. You need to groom assiduously, and have your vet prescribe antihistamines. However, this is so rare, if your dog is displaying symptoms see your vet.
Bacteria on the dogs skin can cause an allergic reaction. Generally they are not allergic to their fur. Many times allergies can be controlled by diet. Try feeding a premium food formulated for "skin and coat". Do not feed kibble or treats that are colored (have dye). If adjusting food does not help see your vet.
Yes, dogs can be allergic to their own fur and dander. It is a hard allergy to deal with, but steps can be taken to minimize the dog's allergies. Regular grooming to remove excess dander and fur helps. Vets can also prescribe anti-histamines to help keep the dog more comfortable.
