Very odd indeed, can't stop brake lights from being on with no switch, my bet someone has seriously been messing with that car, maybe a 'field' car. Poster location says it all.
You simply MUST have a switch of some sort in there, how every brakelight on the planet works on/off. Must have been one at one time, on second OP pic the brake pedal plate clearly shows what looks like a contact mark from the now not present switch stem.
Could be a hydraulic switch like mentioned but I think not.