The wording seems to be in a gothic script variation of what is known as ‘Schwabacher’ which was the most frequently-used lettering style in olden times in Germany. Although displaced in the 1500s by ‘Fraktur’ it remained in use as a secondary style for Fraktur where emphasis was needed and was used in a similar way to italics through to the mid-20th Century. Both styles were officially outlawed in 1941 on the orders of Martin Bormann (after discussion with Hitler), having been declared “Schwabacher Judenlettern” (Jew-letters of Schwabach) with no German heritage.
I think you can be sure your stein is pre-1941 and might well be early 1900s (see note on ‘stein’ at the bottom).
They’re very difficult to read, but I read the second word as “Hudenken”. Although that might be a surname/company name it’s a pretty uncommon one. As an ordinary word it would translate as “thinking”. The first word is more troublesome. It might be “Bunt”, which would translate as “colourful” or “gay” (in a non-LGBT sense) or “festive” in a celebratory sense.
My guess is that this is not a name but a drinking toast with a literal meaning centred around happy thoughts that more liberally translates as “cheers” and perhaps specifically in connection to a beer festival in an Oktoberfest sense.
The same ‘Hudenken’ word appears on other German drinking vessels from around the early 1900s. Here it is on the lid of another stein (at the very top) and also on a teacup. The lettering style here is easier to read, but on these items the word ‘Bunt’ (or whatever) isn’t present. I think it still has the same kind of meaning but without ‘festive’ overtones.
[NOTE: ‘Stein’ simply means ‘stone’ in German and was rarely used to describe beer mugs, which were more commonly referred to by terms such as krug, humpen or seidel. Stoneware/earthenware as a material traditionally used for beer mugs and other containers is known as steinzeug/steingut and a stoneware jug or mug as a steinkrug. The use of ‘stein’ to describe beer mugs probably arose as a contraction of those terms by non-German speakers, later popularised by American soldiers.
The exception is the use of the term ‘stein’ in the very western Rhineland area of Germany, where it specifically refers to a one litre glass beer mug. Glass wasn’t widely used for everyday beer mugs until after it was introduced at the 1892 Oktoberfest.]
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