The real stuff is made by burning plant material and mixing the ashes directly into oils such as coconut and palm. The lye (and the black color) comes from the plant ashes much like the way people used to make lye by leaching it from wood ashes. This results in a very non-specific lye measurement so you COULD end up with lye heavy soap. To prevent this you might overcompensate by adding much more oil than you think you need. And if that oil happens to be something like coconut then it would be drying. Other oils could make it prone to spoilage.
There are also a variety of other soaps that people simply color black using various dyes and call them African Black soap... but technically they are not. That's mainly just label appeal.