The Demon represents King Shahrayar. The demon is set on punishing someone because of his imprisonment by Solomon. This correlates with the King wanting to punish his future wives based upon the action of his first wife.
The fisherman represents Shahrazad. Both are clever and use stories and wit to halt their deaths.
The enchanted lake and cave represent the way that the King is imprisoned in his own anger and need for revenge. The prince that is half man and half rock represents the king locked into his own anger and is unable to move forward in life because he is so weighed down by the anger.
I was also real interested in trying to find out how the number four played into everything. Four times to cast the net, four fish, 400 payment….. I would appreciate anyone’s thought on this.