If Alice, who owns an aircraft and may or may not be a certificated pilot, flies in her plane as a passenger with her friend Bob as PIC, what authority, if any, does Alice have over the flight that an ordinary passenger would not?
They have full authority over their own personal relationship with the pilot. For example, if the pilot does something they don't like they can choose to never let the pilot fly their airplane again. Or, if they have hired the pilot, they can fire the pilot.
However, in the eyes of the FAA they have no authority over the conduct of the flight whatsoever. If the owner persuaded the PIC to do something against regulations in a non-emergency under threat of firing, it isn't the owner who will earn a flight violation in their record... it is the PIC.
This isn't a matter for the regulations. The PIC and owner of the plane should have a private agreement on how the airplane will be used. This can vary from no authority if the PIC has leased the aircraft from the owner with no restrictions on its use to a great deal if the PIC is hired by the owner to transport them somewhere.