A mass spectrometer measures the mass of a charged particle travelling at known speed by noting the distance it moves in a circular arc, given by r = mv/qB for magnetic field B.
The mass of the sun is measured by applying Kepler's Law M = (4 pi^2 R^3)/(GT^2) in terms of the period T of the Earth's orbit and the mean distance R of the Earth to the sun. Observe that Kepler's law is essentially the same as Riofrio's universal law rescaled by 4 pi^2. The value of R itself is now known fairly accurately, but the mass of the sun also depends on an accurate value for the gravitational constant G.
Observe that in all of the above settings the measurement of mass actually depends on the measurement of other quantities, most noticeably distances. In gravity a mass has a characteristic Schwarzschild radius for which r = 2Gm/c^2. Similarly, the quantum mechanical Compton wavelength is given by l = h/mc, but this indicates an inverse relation between mass and length. As is well known, these lengths agree at the Planck mass. This suggests that any theory attempting to explain the meaning of quantised mass ought to incorporate some form of T-duality, a correspondence between lengths and their inverses, up to a scale factor.