Chewing really hard gum -- think super bubble or bazooka -- is one of the tools used by my daughter's OT for sensory issues, as it helps with the mouth muscles and helps organize.
As others have mentioned, the chewing or mouthing of non-food items, as well as the chewing on shirt colors, is a symptom associated with a sensory processing disorder. See if you can get an evaluation with an Occupational Therapist trained in sensory processing disorders.
There are various levels of acceptance and understanding of sensory processing disorder among professionals, and it may not be recognized by the schools and pediatrician or validated by them. The AAP likes for sensory processing to carry an addtional diagnosis, so your doctor may push to find an additional diagnosis. My own experience has been that they are really stretching to find a way to make my daughter fit another diagnosis that they can't seem to make fit without stretching the diagnostic criteria and definition of the other diagnosis which to me means that they're struggling to find an additiona diagnosis that is likely inaccurate diagnosis rather than accept the sensory diagnosis confirmed through at least six different recognized tests and that she fits 100% without having to stretch or struggle to recognize is an issue. The reason they like an additional diagnosis is that the AAP has not agreed on standard diagnostic measures for Sensory Processing. It is not that it cannot exist on its own as much as it is that the American Academy of Pediatrics isn't comfortable with that. So, another diagnosis may or may not be warranted, but they will definitely push to try to make something fit. At the same time, if another diagnosis can be added, you will likely get greater assistance with resources than if one cannot, as insurance likes to have an additional diagnosis before paying for private therapy and as the schools like another diagnosis to support providing their services. So, he may have several issues or it may just be sensory. You'd have to have him evaluated to know.