How would it make any sense to use dimensionless numbers to represent physical things?

From wikipedia:

If you are comparing the strength of forces, and you are using these numbers to do so, I would have thought that these numbers would represent units of - well, force.

The size of any constant with units is ambiguous. Is g=980 cm/sec^2 large or 0.0098 km/sec^2 small?
Expressed as a dimensionless number in 'natural' units, the electric charge of the electron is e^2=1/137 regardless of what units were used before it was made dimensionless, so the EM interaction is seen to be small.