The Complaint Regarding Wash Day/Dish Pan Red Hands
Predates "detergents" by generations.
As one has been saying elsewhere all soaps are alkaine in solution with water, some more harsher (highly basic) than others. Hot water increases the power of soap but it also helps contributing to the drying of skin immersed in such solutions. It was the removal of protective oils combined with high pH that literally destroyed the hands of women or anyone else charged with doing the washing up/cleaning. To add insult to injury sodium carbonate/washing soda came along and that kicked up things a notch.
It was possible to make mild soaps that didn't cause so much havoc with milady's hands, but they often didn't clean as well and were therefore charged with danity laundry and or light household cleaning. Eventually soaps for household use broke down along the lines of heavy duty (Fels, Kirkman's, Octogon, Sunlight), and light cleaning (Lux, Ivory, P&G White Soap and the rest of white or lighter coloured soaps).
The first detergents could be used for any sort of general cleaning as well as laundry, one only has to read vintage boxes to find instructions. This trend continued right up through the 1970's or so when the basic formula's of surfactants, water softeners et al were juiced up with enzymes, OBA's, bleaches and the rest of the chemical cocktails one finds in modern laundry products. That being said there is nothing stopping anyone from using "Tide" or any other laundry detergent for dishes, just make sure to rinse well afterwards.
One important discovery was sodium lauryl sulfate and it's cousins on which "light duty" detergents and later personal cleaning products were based. Dreft, Vel, Trend, Woolite and the lot were all based upon these new "miracle" chemicals as would the new invention to follow, dishwashing liquid.
The beauty of SLS based detergents is that one can blend other surfactants (cationic) and or chemicals to offset the drying nature of the fomer whilst also imparting skin moisturising properties. This is where you begin to see the separation of manual dishwashing from laundry products.
Back in the day when one or perhaps two types of soap did 99% of the cleaning around the home (including personal hygiene)it would take heavy lifting to get Madame to purchase another product just for washing dishes. Same was true after laundry detergents came along but as lifestyles changed so did habits. While extremely alkaline detergents are great or attacking grease laden dishes (which is why automatic dishwasher detergents are still rather caustic), for most handwashing that sort of brute chemical force just isn't required.
Have a stash of Ecolab commercial detergent one often uses for general house cleaning. As the stuff is loaded with phosphates and washing soda it does great job of removing grease/oils and keeping dirt suspended in the water. I've used this product for everything from cleaning the filters on the air cleaner, washing window screens to scrubbing down dirty appliances.
