What other units can influence the way a wire glows than watt?
Power is most likely not fixed.  From your description, it seems that power is at least one of the parameters that is altered as a result of the soft/medium/hard setting.

The heater is most likely a nichrome element.  This is often used for electrical heating since it has relatively high resistivity and can handle glowing temperatures.

The power dissipated by a resistor is proportional to the square of the voltage and the square of the current.  There is probably a switching power supply in your device.  If this were a laboratory instrument, a processor would probably continually measure the voltage and current, then adjust the pulses duty cycle to regulate the power.  In a cheap consumer device like this, maybe it regulates the voltage.  It might just be pre-set to three different PWM duty cycles.