I'll put my chemistry hat on and say that it's possible. How far down the ingredients list is it listed?
Essentially, the soap's formulation is salty water with surfactants (plus a few odds-and-ends). The faucet is chrome-plated (I assume). If there are any defects with the plating that allow the unlying brass to come into contact with the soap then the presence of the chloride will facilitate galvanic action between the brass (mainly copper, of course) and the chromium. The latter will corrode. The presence of the surfactants will lower the surface tension and make it much easier for the soap product to wet the metal surfaces. Also, over many months, the water will evaporate leading to a more concentrated salt solution, too.