Even with the engine off, the car has an "accessory" mode, in which it lights up all visuals on the dash, including oil check engine and many other things. I wouldnt be concerned unless you have problems out of the brakes.
Your brake light goes on because it receives power directly from you car's battery. Brake lights are wired to your battery and have a switch attached right behind your brake pedal that turn them on and off when you press or release the brake pedal. This function is independent from the engine's operation.
The brake lights work anytime you apply the brakes. However it it stays on after you move your foot you need to adjust or replace or brake light switch.
The brake light can come on because brake light wiring. The wiring is hooked
on a totaly different wiring scheme than your engine uses. The brake light
could be made to work simply by running a wire from the break light
to a button put the button on your brake pedal and it will light when your foot touches botton.
