Mathematically, the Bel is unitless because it is a function applied to a ratio of measurements. Whatever units those measurements have disappear due to the cancelation.

If that measurement is linked to another one in a nonlinear way, that nonlinearity doesn't disappear: the fact that if the unitless fraction in which certain units canceled out is 2, then the corresponding unitless fraction obtained via different units is 4.

Just because the units disappeared in a fraction doesn't mean they are not relevant.