Surfactant types and levels are different from different ingredients. That's what does the "cleansing" you're asking about. Not all oils and fats clean the same way. Some are better for different "messes" than others. Like wood ash makes a fantastic soap to strip grease. However, it also dries the hell out of your skin and can lead to lots of cracking.
It's all a balancing act as to how strong of a soap you want compared to damaging.
My understanding was that, when companies add lathering agents to their products, it's the same as dyes and fragrances: really just there for cosmetic purposes.