Is it normal to go see a breast specialist due to mastitis?

Define specialist. If you mean seeing your obgyn or pcp, most can handle your average mastitis. If you mean a general surgeon or plastic surgeon, i'd say no. They could help, but i think they would wonder why you didn't see the other docs.
Yes, necessary. Mastitis can be a recurrent, chronic problem. For a mastitis that clear with antibiotics, then your gyn or pcp can handle. If your mastitis does not clear or it keeps coming back, then a breast surgeon or general surgeon is necessary. You may require drainage of an abscess. Surgery may be needed to remove the tissue causing the mastitis. See a surgeon for best results.