A soap label carries more information than a shampoo label, if you know the vocabulary.
Saponified oils
This tells you what the bar was actually made from. Saponified coconut oil lathers, saponified olive oil conditions, saponified castor oil stabilises the foam. If a bar lists only one, it will be either drying or soft.
Glycerin
Cold-process soap makes its own glycerin and keeps it. Commercial bars often strip it out to sell separately, then add synthetic humectants back. Ours stays in the bar.
What to be wary of
The word ‘fragrance’ can hide dozens of undeclared compounds. If a natural brand will not name what makes the bar smell the way it does, ask.