It doesn't stay on. I have one plugged into the socket where the ashtray would be and got a little concerned about what appeared to be a permanent power supply. It even stays on if you get out of the car and lock it.
However, it appears it is timed. I made a point of checking the other day. Left the little cubby open, got out of the car, locked it, green light still on. Went for a coffee and then when I came back I looked through the window and the light was off. Unlock the car and it comes on again straight away.