Soap is technically edible if you start with edible fats and don't add non-edible things like fragrances. At one time (1700s and 1800s for sure and maybe earlier), pure soap was mixed with medicine to make pills for people to swallow. OTOH, soap has also been used to make enema solutions, but the added soap can cause painful cramping. And if you eat enough soap, it can cause digestive upset.
I can't quite figure out why you want "edible soap" for dishwashing. So you don't have to rinse? If so, that's not a good idea. That last rinse of clean water flushes away traces of food particles and microorganisms that might later cause illness. There's a reason why restaurants use 3 sinks full of water for cleaning dishes -- wash, rinse, and sanitize.
Soap also doesn't work well for washing dishes if you just put it into a sink full of water and dirty dishes. The soap reacts with hard water minerals in the water to make soap scum. When soap becomes soap scum, it doesn't work for cleaning. Soap works best if used directly on a sponge or washcloth to scrub directly on a dish, so the amount of hard water minerals the soap has to deal with is as low as possible.
