We drink a lot of tea and coffee and all of our mugs get terribly stained. I homebrew beer and use Pro Brewery Wash (PBW) to clean up after a brew day. I discovered that it also works wonders on stained mugs.

To clean coffee and tea stains on mugs (NOT teeth) put a tablespoon of PBW in half a gallon of hot water. Not boiling, only like as hot as your hands can stand. Soak the mug for 15-30 minutes and the stain wipes right off. It has never damaged any of the printing on the mugs.

https://fivestarchemicals.com/pbw-cleaner-1bs

You can get it at any homebrew shop or online.