IMHO A stein is better than a mug, a mug is usually what you use for coffee.
The word stein is a shortened version of the German word Steingut [Stoneware], they can hold anything from half a litre to A-Lot-More.
The stein is also well known because back in the 13th century [I think?] German law required that all public places that served drinks had to have lids because of the massive quantity of flies that were plaguing the region.
In modern times this practice has largely been neglected . . . the stein for most intensive purposes has been replaced with the humble glass beer mug [found in most pubs]