Menthol is taken by many people in cough drops but methol exacerbates mucus production ie increases it. Use some alternative and likely containing zinc.

Any irritant or narrowing of the air passage triggers a cough response and here the narrowing is caused by thickened mucus. Eventually the color will become clear, but constant irritation can create a "dry non-productive cough".

Expectorants loosen mucus plugs. Other cough syrup sooth the irritation.

People who are very sick will often swallow their phlegm. What happens often is they started with a viral infection and started swallowing their phlegm (often done by kids and the elderly or disoriented very sick adults). Then food particles are in this phlegm and a bacterial infection happens on top of the viral infection. Their fever spikes, their mucus turns yellow and is thickened. That is when antibiotics are needed.