I always figured McCoy had scientific responsibilities on the Enterprise beyond Chief Medical Officer, which is why he wore the science division badge. Neither way would be "wrong" just interested in how people see McCoy. Has he always had responsibilities outside of medical duties or is it a new thing when he joins the ship.*
*We know that sometime after he was promoted to Lieutenant Commander he served time on Capella in some form. While it was just footage from the episode that we see in the Briefing Room, it showed he already had a bigger part than purely medical.