beer mug vs stein

beer mug and stein both are nouns.

Word	Noun	Adjective	Verb	Adverb
beer mug	Yes	No	No	No
stein	Yes	No	No	No

As nouns, beer mug and stein are synonyms defined as:

beer mug (noun)	stein (noun)
a mug intended for serving beer	a mug intended for serving beer