Allergy is not a disease but a form of the body’s immune system response when exposed to substances called allergens. Allergens make contact with the body in a variety of ways. Allergens can stimulate an allergic response in the body when ingested (eaten), breathed-in, brought into contact with the skin or injected into the body.
Asthma is a disease of the lungs. Persons are born with a tendency to have asthma inflammatory reactions (in the lungs). Sometimes this occurs in response to breathed-in allergens, to infection, to irritants like smoke, and to chemicals. Some people have asthma without an obvious cause. Most but not all persons with asthma have wheezes; some just cough. Some asthma is allergic but many cases are not. You should ask your son’s doctor why be believes allergy to be the cause of his cough. I can’t answer that question.
There are many causes of cough, in persons of all ages. That he has normal sinus and chest X-rays is good news but the “normal” X-rays do not rule out lung or sinus disease.
Please seek further advise from his Doctor regarding other possible causes for your son's cough.
Swallowing Disorders,Gastroesophageal Reflux, Airway Compression (such as a Vascular Ring or Hemangioma) and some infectious or immune disorders such as Immunodeficiency, Tuberculosis, Allergies, Sinusitis, Tonsolitis and Adenoiditis are examples of conditions that can present as cough. Please discuss these with your son's Doctor.