My secret for cleaning mugs after black tea: rinse the mug and shake in some baking soda, enough to be just a bit dry but clump up. Smear it all around the mug and watch all that staining come right up. Itβs amazing how well baking soda cleans so many thingsβ¦
My intuition is baking soda toothpastes clean tooth enamel similarly, but even if you keep swearing off caffeine, this trick can really deep clean any mugs you still have.