A. Coughing isn't a usual symptom of strep throat, which instead more typically causes a sore throat, high fever, and swollen glands. Other strep throat symptoms sometimes include vomiting, abdominal pain, and a rash (scarlet fever).
Children under age 3, although they don't usually get strep as much as older kids, are more likely to get a respiratory tract infection instead of classic strep throat when they become infected with the strep bacteria. Still, you wouldn't expect him to be getting worse after 5 days of antibiotics. Strep throat usually responds very quickly to antibiotics, and many children feel much better within a day or two.
So if it isn't strep causing the cough, then it is possible that he caught a cold virus on top of the strep infection. Has he been around anyone else who has a cough?
Or if he has allergies or asthma, then maybe something triggered an attack.
Another possibility is that he never had strep at all and his symptoms are all from a virus, which would explain why he wouldn't be getting better after 5 days of antibiotics. This is especially likely if he didn't have a strep test and he was simply diagnosed because it 'looked' like strep.
However, even if he had a positive strep test, many children are carriers for strep, which means that the strep bacteria lives in their throat, but isn't causing problems. Strep carriers will be positive when tested for strep, even when the strep bacteria isn't causing their symptoms. This might be likely if he always tests positive for strep, even when he doesn't have typical symptoms or isn't around anyone else with strep.
That leaves us without an easy or definite answer, but since your child isn't getting better and is already being treated by his doctor, you should call to give him or her an update and get further advice.