Saponification is the process of creating the free fatty acids from the triglycerides that carry fatty acids around in the body. Triglycerides are glycerol esters of fatty acids and are the major constituent of bodily fat in animals.
Soap is created by breaking down the triglyceride esters releasing the free fatty acids and is usually accomplished by reacting the fat with sodium or potassium hydroxide. The resulting compounds–the sodium or potassium salts of the free fatty acids–are what we call soap.
Soaps act as surfactants which can solubilise dirt and fat in water. Their primary action as surfactants results from the long chain fatty acids having a polar end that likes water and a non-polar end that likes the fatty and water-insoluble elements in the dirt. It is the fatty acid not the counter-ion that dominates this effect. Swapping the counter-ion (sodium or potassium, etc.) for something else make a small difference to the character of the soap but not a large difference to the surfactant action as that is dominated by the free fatty acid.
So swapping the counterion isn't the point of saponification: the point is to get the salt of the fatty acid free from the triglyceride. And the resulting salts are ionic compounds where the counter ion of the acid salt is not hugely important to the function as a soap.