In both medieval Europe and Song-dynasty China, people would brew alcohol in their homes, beer and rice wine respectively, and then sell the surplus to passersby. The interesting thing is that in both cultures a broom was displayed outside your house to show that you had alcohol for sale.

What might be the source of this shared symbol? Why did this occur both places around the same time?

My source for the chinese side of the story is Water Margin aka The Bandits of the Marsh. My sources for the European side is a couple youtube documentaries, but I could dig up a primary source if needed.