Why is this digital potentiometer working?
My question then is - why is this working? The follow up is do I really need a dual-supply potentiometer or could I get something cheaper?


I can't answer the first part but can help with the second.



Figure 1. Probable innards of OP's dimmer control.

I think your diagnosis of the PSU input is correct and that the PSU control circuit is as shown in Figure 1. Please see my answer to 
0-10V Digital Rheostat for LED Dimmer? for a full explanation of the operation and control of these power supplies using potentiometers, 0 - 10 V control signals or, most appropriate in your case, simple PWM control. (The PWM gets filtered by the low-pass filter R3 and C1 to give the 0 - 10 V control signal.)

Note that with a suitable drive transistor you can drive a very large number of dimmers in parallel. No need for digital or analog pots.