Actually, that isn't fully correct.
Scales do measure weight and with g_moon = g_earth/6 the weight would be 1/6 that measured on Earth. Weight = gravity * mass (i.e., it's a force).
A kilogram is the unit of mass and the mass of a person does not change depending on the local gravity. This person would be 72 kg on Earth, on the Moon, or in orbit.
Regular bathroom scales measure force, or specifically weight. Because we are used to measuring objects in kg, these scales actually report not the measured weight, W, but rather W/g, a mass unit. The 72 kg person weighs 706 Newtons (the force unit) on Earth (with g_earth = 9.8 m/s^2), on the Moon the weight of the same person would be 117 N. If you measured that person with an Earth scale which reports W_moon/g_earth you would indeed get 117 N/ 9.8 m/s^2 = 11.9 "kg".
If you used one of those physicians balance beam scales (the ones with the sliding weights), you are directly measuring mass by balancing two weights against each other. With that scale it would (correctly) read 72 kg on the Moon and on the Earth. (It would not work in orbit as there isn't the gravitational acceleration pulling on both masses to properly balance.)