Primary Ingredient difference between shave soap and hand washing soap?

I understand that there's a difference between soap that has undergone saponification and detergent that isn't soap and hasn't saponified, also soap is a natural product and detergent is synthetic.  Here are some links to more information 

https://byjus.com/chemistry/difference-b...detergent/


https://www.nycoproducts.com/resources/b...detergent/


https://www.quirkyscience.com/difference...detergent/