Not understanding shunt regulator
Could this work without the Zener? .....only if the motor current is constant, which seems quite unlikely.  

Motors draw current related to their load yet in the circuit shown they have 190 mA flowing through the motor and just 5 mA through the Zener.
If the load on the motor caused more current to flow then voltage will drop below 12 V and make it easier to stall the motor.
And if the motor for any reason is disconnected or lightly loaded then more (or all) current will flow through the Zener.

If the motor current varies with a design maximum of 190 mA @ 12 V, then the Zener is needed and could maximally be required to dissipate 2.34 W.  

This seems to be what the circuit is designed for, with a maximum expected motor load current of 190 mA @ 13 V.
If the motor is disconnected, all current (195 mA) flows through the Zener while maintaining 12 V.
If the motor is connected, the load current can range up to 195 mA and still be within design spec at 12 V.

If the load current increases beyond 195 mA due to extra mechanical load on the motor, the voltage will start to fall. Stall (or locked rotor) current for the motor could potentially approach 350 mA (you'd need to know the rotor resistance to calculate accurately) or so with close to 9 W dissipated in the 62 Ohm resistor.