It’s a rule of thumb for cartoons: outdoor bathrooms (in English, outhouses) made of wood almost always have this detail. But the truth is that it is difficult to define the origin of this tradition.
Before understanding the symbol, let’s understand the need for the hole: the houses needed to be lit and ventilated – after all, we are talking about a bathroom. The brand was made at the top precisely to respect the privacy of users.
The most widespread story about the origin of the Moon is that it was used to distinguish male and female bathrooms. In the early 19th century, most people did not know how to read, so the way was to use symbols: Sun or star for men, Moon (associated with the Roman goddess Diana) for women.
In time, only female houses would have remained. The men’s bathrooms were disgusting and poorly maintained (some things never change), and the men gradually started using the bathroom with the moon on the door – and the symbol stuck.
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Not cool? The problem is that this story is probably a joke, and it was invented after the moons were already popular – a myth, to explain how they came about. Especially because, before the 20th century, anyone who lived a difficult life (and with few resources) to the point of building a house would not have as a priority to create one for each sex.
Another problem is that, in many cultures, the sun is a girl and the moon is a boy. Most likely, the Moon, in addition to a nice ornament, would give a good handle, because its concave shape is ideal for fitting your fingers. And then the idea became a meme among artisans.
In the end, what counts is tradition: if the person learned that a outhouse it was made with a moon at the door, that was how the teaching would be passed on. It is also worth saying that, in other countries, the most common symbols are others. In Germany, it is a heart (an attempt, perhaps, to make the task of going to the bathroom more pleasant); in Poland, a diamond.
Sources: Atlas Obscura and Missouri Folklore Society