Having your mouth washed out with soap is a "traditional" punishment, usually inflicted on those that have committed a mouth-related offence like swearing (if you can believe it) or smoking. It's now mostly a figure a speech. The brown soap is a less desirable bar of soap to have forcibly inserted into your mouth.
I think their question was why is it less desirable
Allegorically it's a cheap bar of soap that tastes worse than a more expensive bar of soap that is dyed to not be an ugly colour.
I think it's soap with extra lye. It's more for cleaning floors than bodies.