Hello and welcome to e health forum.
In reactive airway diseases like asthma, the cough is due to reduced diameter of the airway which restricts the air flow.
In conditions, like viral infections induced colds and coughs, inflammation of the respiratory tract and production of mucus, causes irritation of the airway, and hence results in cough.
In young kids, with asthma, superimposed viral infections (especially during winters), worsen the symptoms, and can complicate recovery.
Inhalers contain different type of medications. Bronchodilators (Albuterol) help to improve the breathing, and not necessarily to reduce cough. Steroids (Fluticasone in Advair) reduces the inflammation and hence reduces the hyperreactivity of the airway.
Hence the inhalers your son is being prescribed, are to control his symptoms and not just to reduce cough.
The recovery in such conditions will take a still longer time, which is needed for healing of the respiratory mucosa affected. In the mean time, Inhalers, anti inflammatory medications, and other supportive measures will need to be continued to help control the symptoms.