Could PWM damage the unit?
Only one way to find out!
IMHO, PWM is probably not the best way to temperature control a rice cooker. If it's one of those induction rice cookers, just controlling power input with PWM and adjusting the duty cycle will not work. But if it uses a simple 12V resistive heating element, it should work in theory.
Regardless, if you are thinking of controlling this with something like a 1KHz PWM signal from a micro controller, it's way overkill. The thermal mass of the cooker and food acts as an energy capacitor and doesn't require fast switching frequencies to maintain a constant temperature. There is a reason most cooking appliances are very dumb and use just a basic thermocouple or bimetal thermal switch to cycle the heating element on off over a period of many seconds or even minutes. You can hear this "clicking" on many rangetops and ovens.
If you really want to computer control this DIY style, then I think just a relay and thermocouple tied to a microcontroller will be the most effective and simple option. That being said, if you want better temperature control accuracy and not huge swings in temperature, you will want to implement and tune a PID loop in your controller. If you are not familiar with control theory, it may be a little daunting at first but great stuff to know.
Cheers