The notion of kilogram is also something fuzzy in my opinion and your expression here is revealing... Weighing 1 kilogram... But the kilogram is a unit of mass, not of force, and therefore not a measure of weight. "
The weight of a body is equal to the product of its mass and free-fall acceleration". The IS of force is the Newton, not the kilogram. Less fuzzy.
For a long time no distinction was made between the mass and weight of bodies. Thus, a kilogram served as a unit not only of mass but also of weight (the force of gravity). Differentiation between units of mass and weight was established at the Third General Conference of Weights and Measures (1901). A decision of the conference emphasized that
the weight of a body is equal to the product of its mass and free-fall acceleration and introduced the concept of normal weight and normal gravitational acceleration (980.665 cm/sec2). A separate unit of force and weight—the kilogram-force—was established at that time. The same principle is preserved in the International System of Units, where the newton has been adopted as the unit for measurements of force.
And here is one good definition of what a kilogram is, i.e. a unit of mass, not of weight:
Kilogram
a unit of mass; one of the seven basic units of the International System of Units (SI). It is equal to the mass of the International Prototype Kilogram, kept at the International Bureau of Weights and Measures. Its abbreviated designations are as follows: Russian, kg; international, kg.
Still, the Law of identity is so fuzzy that one kilogram is 0.028 g greater than one kilogram:
In the 18th century, when the metric unit system was first introduced, a kilogram was defined as the mass of 1 cubic decimeter (dm3) of water at 4°C (the temperature of highest density). However, the mass of the prototype kilogram (a cylindrical platinum weight made in 1799)
was found to be about 0.028 g greater than the mass of 1 dm3 of water.
So, a kilogram is not a thing!
and a kilogram is not something that needs to weigh anything!