Torque is proportional to current. Power out = torque x rpm.
At stall (locked rotor) there is no rpm, and therefore no power out. But torque is still proportional to current. Current is determined by the applied voltage and the motor's internal resistance, with the value determined by Ohm's Law (I = V/R).
If the motor is running at some rpm >0 then it also acts as a voltage generator, subtracting from the applied voltage and reducing the voltage across its internal resistance. Therefore to get the same current the input voltage must be higher, and the power input is also higher. This extra input power provides the mechanical output power that is now being produced. Some extra current is also required to overcome 'iron' losses inside the motor (which are proportional to rpm). However it is usually quite low - in this case 1.8A at 5676rpm, which reduces torque by less than 2%.
This is how the same motor can produce the same torque at 'wildly different power values'.
You can see the current is about 25A and the voltage is about 10V.
This gives a power of 250W at stall.
Now, looking at the data sheet, they list a stall torque of 2.6 N-m.
They also give a voltage of 12V and current of 105A. This equates to
power draw of 1260W.
When a controller is used to limit current the motor current may be higher than the power supply current.
The controller uses PWM to lower the effective motor voltage, relying on the motor's winding inductance to smooth out the current flow. Since the controller recirculates current through the motor during PWM 'off' periods, motor current is increased by the same proportion as voltage is reduced. This recirculated current is not seen by the power supply, so the power supply current may appear to be lower than expected.
This explains why the 80A torque test shows an input current of about 25A. The motor current is 3.2 times higher than the power supply current, which also implies that the motor voltage is ~3.2 times lower than the power supply voltage, ie. about 3V.
But what about the '2.6 N-m requires 12V at 105A = 1260W' anomaly? I suspect this is due to an invalid assumption that the empirical stall current is at the 'nominal voltage' of 12V. The motor probably has a lower voltage on it for this spec, just like in the 80A test. If the controller had to apply 33% PWM at 12V to get 105A motor current then the power would be 4V * 105A = 420W.