I have mild asthma and got bronchitis that lasted 3 months. I've been recovering for th last 6 months now and am gradually losing the cough. A cold that goes to your chest can make your lungs hypereactive. Inhaled corticosteroids might help the cough by reducing inflammation. He may have asthma.
Sometimes your lungs get hyper reactive after a cold and the cough can last for months after.
However, a cough can also trigger asthma so best to get it checked out. That is exactly how I started and it got worse and worse until I was waking myself (and everyone else in the house) up 4-6 times a night and was exhausted from coughing so much.