Funny how this question got away from me. I MAKE soap. Soap is saponified oils (fats)...long chain fatty acids, what ever you want to call them.
To make soap from oils (fats..etc..) you make a strong alkali solution (high pH), usually with Sodium Hydroxide, otherwise known as lye. Everything is measured out very accurately and the chemistry tables are checked to make sure the balance is right so that there won’t be too much lye in the mix, ending up in a very harsh soap. In fact, most of us soap makers use a small percent extra of oil in our recipes to make sure the soap is mild and doesn’t strip the skin mantle.
So, when you take lye (NaOH) and add it to long chain fatty acids, it pulls those chains apart and create an entirely new substance. Soap, or soap salt. (chemically speaking, the resulting bar is made up of a soap salt, glycerine and usually some left over water and perhaps some perfume or essential oils and colourant like oxides of some sort.)
The soap salt will have a specific name, depending on the oil (fat etc) used in the mix. For example, I might use Olive oil, so saponified olive oil is called Sodium Olivate. Palm oil is called Sodium Palmate. I use veggie stuff in my soap, but you can see on the shelf at the grocery store stuff made with animal fat. It will have Sodium Tallowate (usually cow fat) listed and sometimes Sodium Lardate (pig fat).
MOST of the liquid soap you find is actually detergent, but some liquid soaps are real soaps and you can tell by reading the ingredient list. Liquid soap is made with Potassium Hydroxide instead of Sodium Hydroxide, so a saponified oil in a liquid soap might be called Potassium Cocoate (coconut fat or oil) it may just say,... the saponified oils of…. and then list the names of the oils used.
I hope this helps, and I realise this is half a year late.