Myth: Zoro is the First Mate of the Straw Hat Pirates.
Fact: Zoro was the first to join Luffy in what later became the Straw Hat Pirate crew, but neither Luffy, Zoro, nor anyone else has ever said Zoro was the first mate. Each member of the crew has had a job profession assigned or announced already in the storyline. To date, Zoro is regarded as the swordsman of the Straw Hat Pirates and a former bounty hunter.
Arguments over Zoro’s classified role have occurred in the past, but to date officially Oda has never listed anyone as the Straw Hat Pirates First Mate and it is for now considered incorrect to name him as the First Mate until stated otherwise.
Furthermore, to Luffy all the crew members are on the same level, he views them all the same, so he’d never say “Zoro is above the rest of you in ranking”. Zoro also wouldn’t really care about having some meaningless rank above his crewmates/friends. The only title that Zoro wants is that of “Greatest Swordsman in the World”.
If the crew needs a voice of reason he’ll be there and play the role in spirit which is all that really matters. He knows he can direct their attention and be the voice so he doesn’t need some reminder.