So there is no module in the underhood fuse box or the large box next to the battery? Keep in mind that all of the connections needed for a remote start can also be found under the hood - it is not 100% required that the remote start be installed near the ignition switch - altho that is where most put them.

You have a bridge between battery feed and switched ignition that is interrupted by pressing the brake pedal - sounds like a remote start or pushbutton start kit to me. Of course, ANY bridge between battery hot and switched ignition would explain this - but the fact that pressing the brakes shuts the car off certainly implies a remote- or pushbutton-start.