It's better to be safe than sorry I would take him in to an eye doctor, and what kind of drops did they give you? For allergies I would think he would be on allergy meds or even nose spray not drops. If his eyes check out OK from the eye doctor, you might want to think about taking him to an allergy doctor.
Does he complain they hurt, or itch? do you notice him squinting?
My youngest got glasses at 18 months and my oldest has allergies and started on medicine fairly young. I would take him in to both doctors. Our pediatrician did not catch my daughters eyes, I did and she is very close to being legally blind. Obviously she is no longer our doctor, there were other issues as well.