well......despite the fact that both mouthpiece of a pipe and some saxophone mouthpieces are both made or ebonite, there is a difference which is removing any material from the tip, rails and the chamber of a mouthpiece will alter the mouthpiece performance.
So, if the best advise is don’t let your mouthpiece ever get to that point, but if you need cleaning it that much ...you probably should do that only with the parts that are not involved with sound production.
This kit seems to be working well (although I imagine it takes a little more than a good rub with both components).
Component 1, says the ad, is sanitizing the mouthpiece but clearly has to contain some agent apt to dissolve some of the offending dirt and removes scratches by means of abrasion
Component 2 is only wax.
I would advise caution in the use of component 1.