My 30" gelding tends to cough after he has whinnied. His winny if you can call it that (more like a goat being throttled and strangled) is mostly air then he coughs about three times. Never when eating or drinking.
I have asthma, when its bad I can't shout or sing etc because it makes me catch my breath and cough. I suspect your horse has a breathing problem. Have you tried him on a herbal breathing supplement? I find Clarity from Feedmark is good or Breathe Free from Dodson & Horrell or Easy Breathing from NAF. There are others available but I haven't tried some and others I didnt find effective for my horse.
I wouldn't touch naf tbh. Load of tosh. But maybe I can look into adding some mint or eucalyptus? May clear him up? I have said before that he breathes a little funny.