Softens Your Hands While You Do The Dishes
People have been using dish washing liquid to wash hands for decades now. Ever since the first light duty powders (Trend, Dreft, etc...) were replaced with washing up liquids that became far milder.
Original light duty detergents were powerful anionic surfactants that were great for dissolving grease, but also one's hands as well if used long/often enough. Later products used various less harsh detergents that still got job done, but again being less rough on skin.
Of course there are all sorts of washing up liquids ranging from quite mild to very harsh. Same as with shampoo really. Prell is known for what it has been almost since beginning, a very strong shampoo. Dawn when first introduced was powerful on grease, but you wouldn't want to wash your hands with it on a routine basis.
Ivory Liquid practically cornered the gentle dish washing liquid market. Helped along no doubt by association with Ivory soap and Ivory Snow.
Ivory liquid was long used in many households as a hand soap, laundering delicate things by hand, and most famously in some communities as a bubble bath.
Peach Thrill (Mama's brand) was another dish washing liquid marketed for mildness that found other duties as well.
Dove was another brand that relied on association marketing (Dove is one quarter cleansing cream, it simply can't dry your skin like soap).
Finally there was Lux with "Dermasil".
https://www.youtube.com/watch?v=QaxPFuu7ZnQ
Obviously sooner or later clever chops at various companies were going to put two and two together, realized there was a market for liquid "hand soap", so race was off.
If you examine ingredients closely there isn't much difference between mildest liquid hand soaps compared to washing up liquids that are equally gentle. The latter besides using perhaps more gentle detergents, buffering the pH a bit may also add moisturizers and so forth.