The crew addresses her as Captain Georgiou several times in Such Sweet Sorrow but she has no uniform. Why is she still wearing s&m gear and being addressed as "captain" on the bridge of the Discovery?
Mirror!Georgiou negotiates with Sarek the terms of her freedom in "The war without, the war within":
GEORGIOU: You face annihilation. Is it not logical to do anything you can to save the lives of your kind? Give me what I want, and I can assure victory for the Federation.
SAREK: What are your terms?
As a result, Mirror!Georgiou is now posing as the real one, as seen in the last scene of that episode. The speech given by Admiral Cornwell strongly implies that this move is a desperate measure taken by Starfleet:
CORNWELL: Open a shipwide channel. We have all mourned the enormous loss of life due to this war. The acts of violence committed against us are the acts of a foe without reason, without honour. And they will not stop coming after us in the hopes of destroying everything that we hold dear. These are desperate times, and they call on us to do more than merely protect our people, defend our borders. (...) . So allow me to introduce you to the person who will chart your course to Qo'noS. Captain Philippa Georgiou.
GEORGIOU: Thank you.
CORNWELL: Though long presumed dead, Captain Georgiou was recently rescued in a highly classified raid of a Klingon prison vessel. She was transported aboard Discovery with my personal retinue. The mission to Qo'noS is a perilous expedition, and Starfleet is confident that Captain Georgiou is uniquely qualified to get you there and to do what needs to be done.
GEORGIOU: Specialist Burnham, I trust that this will be an auspicious reunion, despite our history. Now take your station.
BURNHAM: Yes, Captain.
So in short: Mirror!Georgiou is posing as the real Georgiou, who is still a member of Star Fleet.