I didn't compare the part number, but that looks like the trailer electrical module. It plugs into the vehicle wiring harness, and converts the multiplexed vehicle wiring signals into left, right, brake, and tail light power. The harness runs from it, to the plug in receptacle on the hitch drawbar.

I am a little confused how you can install everything else without installing this, since everything else connects to this.

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2007 X3 3.0si, 6 MT, Premium, White

Retired:
2008 535i, 6 MT, M Sport, Premium, Space Grey
2003 X5 3.0 Steptronic, Premium, Titanium Silver
2002 325xi 5 MT, Steel Grey
2004 Z4 3.0 Premium, Sport, SMG, Maldives Blue