
Ozma's style of government seems to be pretty informal. There is a noble class in the Emerald City, but it might well be a remnant of earlier times. Wizard refers to "many ladies and gentlemen of the court, all dressed in rich costumes. These people had nothing to do but talk to each other, but they always came to wait outside the Throne Room every morning, although they were never permitted to see Oz." Ojo's trial in Patchwork Girl is attended by "many of the nobility of the Emerald City, lords and ladies in beautiful costumes, and officials of the kingdom in the royal uniforms of Oz." And Rundelstone reports that Ozma routinely has breakfast with her ladies-in-waiting. Ozma has actually promoted some people to the nobility herself, having made Dorothy a princess, Jenny Jump a duchess, and Captain Samuel Salt a knight. One office Ozma apparently didn't retain was that of Prime Minister; her father had one, but there's no indication that she does.
What the fairy queen DOES have is a council of trusted friends who advise her. Glinda lists the members of her council as the Scarecrow, the Tin Woodman, the Patchwork Girl, the Shaggy Man, Tik-Tok, Jack Pumpkinhead, Cap'n Bill, Professor Wogglebug, the Frogman, Uncle Henry, and the Wizard of Oz. The Scarecrow is identified as Chief Counselor in Ruth Plumly Thompson's unfinished short story "The Enchanted Tree of Oz," and that makes sense to me. Some of the others also have official titles: Shaggy is the Keeper of the Royal Storehouses, the Wogglebug is Public Educator, and the Wizard is mostly known by his title. The fact that Henry is mentioned as "[t]he best farmer in all Oz" suggests that he might be sort of an unofficial Minister/Secretary of Agriculture. I also think it's likely that some newcomers to Ozma's court in the post-Baum books have joined the council. Herby the Medicine Man, for instance, probably has some say in medical matters in his position as Court Physician. And maybe the Doubtful Dromedary is the Royal Devil's Advocate, or something like that.