Why do many old shaving mugs have the words "Remember me" or "Think of me"? No name or picture to remember, just those words. I think the German version is "Zum Andenken" translated as "in memory of".
Not completely sure but I think it has to do with the fact that in those days, men didn't shave at home but were shaved by the barber. A man had the option to purchase a mug and brush for his own personal use, which the barber stored on a shelf for him. It was a tradition that when a customer died, the barber would present the mug to his family as a memento.
Or maybe he didn't. The barber wouldn't have any further use for the mug, and turning it over to the owner's family would make sense. The Internet has a lot of information of varying degrees of accuracy. This is especially true for auction sites. Many "antique" mugs were actually made as souvenirs in the 1950s - 1970s. "Remember me" might well have been inscribed on gift items. Someone at the The National Organization For Barber Shop should be able to provide a definitive answer.
If you've ever read the book A Tree Grows in Brooklyn, one of the passages describes the family stopping by the barber shop after the father dies, to get his mug. The barber washes it out, and gives it to the boy, and tells him what a good man his father was. Even though he was really an alcoholic bum.