A coworker tipped me to that many years ago, for my glass-lined coffee thermos. It actually doesn't take much, just a tbsp powdered dishwasher detergent for a pint thermos is ample, and a 5minute soak using hot tap water will do the trick.
I appreciate you mentioning it. I needed to serve half n half in a stainless coffee carafe to a huge crowd recently and fretted over the stains inside for two days. Never did get it bright; no one saw it, but I'd have felt much better if it was not stained.