The mass of a biological molecule is evaluated by the process of Mass Spectrometry (specifically MS/MS or MALDI-TOF). These are an improved version of Mass Spectrometry(MS) and results from the works of physical chemists.In a nutshell, you can ionize the molecule. It dissociates into several ionized particles. Then subject them to a magnetic field and an electrical field at the same time.
The ionized particles take a curved route while moving , and the amount of curve depends on m/q of molecule ( mass of particle (in kg) and charge of it)
Molecular weight is dimensionless indeed. However one may add Dalton/amu/u "False" physical units for better understanding.
The best other example in physics is radians. Angles are defined in radians in physics. But they are dimensionless.
In Circular motion , we have these formula :
v = w * r
teta = w * t
unit of v is (meter per second) and unit of r is (meter). therefore the unit of w (omega) is s^(-1) .
unit of w is " s^(-1) " while the unit of t is "s".
So teta is dimensionless although we describe it as radians for better understanding