Why and how am I supposed to change the extrusion multiplier? The slicing software knows exactly (given accurate E-step, filament diameter, nozzle diameter values) how much cold plastic is pushed in the extruder and how much mass is extruded.

Why should the extrusion multiplier be changed from 100%? Shouldn't the slicer extrude already enough to fill the gaps left by the neighbouring perimeter (see rectangular vs recto-oval extrusion profile).

As result of the "why" question, implicitly comes up the question "how" should I (scientifically, not try and fail) define the extrusion multiplier? it would appear that said value should not be touched if the rest is done properly.

The only idea I can have is to print at 100% infill and then saw the part, to check how big are the voids between 4 neighbouring extrusions. Measuring the wall thickness seems very prone to errors (but again, is such calibration really needed?)
In an ideal world, you will not need to change this parameter once it is properly calibrated.

In a non-ideal world, some filament may be out-of-spec, and some filament may slip when it passes through the extruder. So for a flex (or more flexible) filament, you might increase the extrusion multiplier a little to compensate. If the compensation is material specific, it should be consistent (and might even be advised by the filament manufacturer).

You might want to increase/reduce the multiplier whilst printing the first layer (in lieu of fixing your bed leveling properly).

Maybe you want to fine-tune the top surface (which will be 100% fill) and you prefer to slightly under-fill this (because over-fill results in more noticeable surface defects).

It is for quick hacks or fine tuning, there is probably no scientific approach to setting any value other than 100%.

When it comes to material specific variations (due to melt viscosity, thermal expansion or drive efficiency), there are many factors which affect the tuning. Machine geometry, temperatures, colourants (and other additives in the filament), as well as the base material.