Speaking in a technical sense, soap -- the stuff we like to make from fat and lye -- is indeed a surfactant (SURFace ACTive agent). A surfactant is something that allows other chemicals to mix together that otherwise would not mix. Soap is good at convincing water and fats to mix.
More specifically soap is a detergent -- a surfactant that can be used as a cleaning agent. I see many sources make a distinction between soap as being something different than detergent, so this is a bit of a gray area. IMO, it's more accurate to distinguish soap from synthetic detergents (syndets).
Soap is not a synthetic detergent because it can be made in nature as well as by humans. A syndet is one that is cannot be created by processes found in the natural environment.
I know NaOH isn't a chemical found in the natural environment either, but soap can be created by natural processes using alkalis that are found in nature -- sodium carbonate, being one example.
Misschief is correct that sodium olivate is plain, nothing-special olive oil soap. Sodium olivate is the INCI term for this type of soap.