This is likely more subjectively true than objectively and is relative. When you have the virus the cough doesn’t seem that bad because you have several other symptoms. Then once the virus clears and you’re stuck with mostly just a cough it’s the only thing your brain is focused on. It’s also worth noting coughing begets coughing, so using cough drops to reduce cough frequency will help reduce severity of cough after the effect of the cough drop wears off.

Pharmacist here - antibiotic generally does refer to antibacterial agents, with terms antifungal and antiviral used for those agents. But the word antibiotic technically could be applied to all of the above.