In grade school, we learn that things on the Moon weigh about 1/6 of what they weigh on Earth. In highschool, we learn that the gravitational force an object exerts is proportional to its mass. At some point in time, I must have combined these facts into the idea that the Moon’s mass is about 1/6 of the Earth’s. I’ve heard someone else say the same thing, so it is probably a common mistake.
What’s interesting is that I don’t remember ever thinking about the mass of the Moon. I didn’t misunderstand anything either. The wrong fact just ended up in my head somehow. I only noticed the wrong fact because I thought it’s kind of weird to say “the Moon goes around the Earth” if they revolve around their common center of mass. (It’s not weird, because their common center of mass is in fact inside the Earth.)
The surface gravity of a spherical object is not only related to its mass, but to its radius as well. The Moon’s mass is about 1/80 Earth masses but its radius is about 2/7 Earth radii, so by Newton’s law of gravity, the Moon has a surface gravity of about (7/2)2/80≈1/6 Earth’s.