Aurora R7, does the liquid cooler use standard mounting?

I plan on using the same CPU (I7-8700) so I shouldn't have a problem moving the liquid cooler to another LGA 1151 300 series motherboard?

I doubt you'd be able to reuse the Alienware liquid cooler on most aftermarket cases, unless it's a SFF case. The reason is unrelated to the mounting system (although you'd have to unglue the mounting backplate from the motherboard).  The reason you'll have a problem is because the liquid tubes are specifically cut to length for the aurora r7 case, and are only about 1/5 the length of your standard AIO water cooler. So you will probably have trouble attaching the radiator to the vent in an aftermarket case, considering size limitations.  Otherwise, yes it's 1151 standard Intel 75 mm spacing for the mounting bracket on the AIO cooler. 

The fans in the R7 are standard 120mm 4 pin pwm fans, standard wiring, so yes you can use them in another aftermarket case, but they aren't especially high quality fans (you'll get a lot of noise from them at high RPMs)/ 

Consider a Corsair H60 or Corsair H75 as a comparable retail equivalent to the Alienware AIO if you want a simple AIO to replace it.