Yeah, it’s an officer vs enlisted thing.  In modern militaries, no matter how high ranking an enlisted man is, he’s still considered lower-ranking than any officer, and Lieutenant junior grade (Bashir’s rank) is an officer rank, albeit a low one (it’s one step up from Ensign, the lowest.  Bashir’s a j.g. instead of an Ensign because of all the extra education he got in medical school.  The US Navy operates on similar principles.)