It looks like a 3 pin electrical connector so it might be a connection to a pump or something that needs to operate at a variable rate but I have never seen anything plugged into it on any RHD JK's so it's most likely unrelated to your issues. Mine has never been used - I can tell by all the dirt still caked onto it
Out of interest, does the brake pedal feel firm and unresponsive directly after startup when you unplug the power for the vacuum pump? Or does the pedal feel the same, regardless of vacuum pump assistance?