First, I am not a rubber chemist, but here's my understanding:
The brown stuff is actually rust colored, because it is an iron based antioxidant (AO). These AO's are in both the tire and some of the tire shine products. They don't harm anything - other than looks.
Some interesting facts:
None of the tire manufacturers makes, markets, endorses, etc, ANY tire shine product. Michelin did for a short time, but they no longer do. I wonder why. Does legal liability have anything to do with that?
I am aware of several tire shine products that are actually harmful to tires. They strip the protective AO's and waxes off the sidewall there by accelerating the aging process. Typically, these contain petroleum distillates.