How can I measure a TIP120's beta?
The first step is of course to consult the datasheet and what do we see there:



Some resistors across the BE junctions!

Let's assume that the Vbe of the input transistor is 0.5 V (the current is quite small) then 62.5 uA will flow through R1. I don't know what the the voltage at the emitter will be but let's assume it is 7 V as a starting point, then 7 V / 51 k = 137 uA flows. Those values are very close to each other so the current through R1 cannot be ignored when measuring beta.

So the base current you measure is likely not the base current of first the internal transistor.

Also note the conditions stated in the datasheet for which the beta (or Hfe), with a value of 1000 is listed: for Ic = 0.5 A or 3 A and Vce = 3 V. Are you meeting those conditions? I think not due to your very large (51 kohm, is that correct or a typo?) emitter resistor.