You were very close when parsing the sentence but the preposition(-like) is actually “tout en”, instead of “tout” and “en” as separate words, and means “while”, “at the same time”, “as well as”.
In this particular sentence, “en” used alone could be misleading because it could both mean “by” (expressing the way by which that cream moisturizes the skin) or “while” (expressing the fact that the cream moisturizes the skin AND, at the same time, cleans it).
“Tout en”, by contrast, is unambiguous.
1) for your sentence, “tout” means “and/while”, let me explain:
Life Brand hydrating hand soap with pure aloe extract hydrates skin in a manner/style of softness AND/WHILE cleaning and leaving hands soft.
2) linguee.fr website should have helped you for such need: