what is the mechanism below the shifter in automatic cars called?

shifter linkage!

you should verify the independent secure practices swap, the brake swap and the linkage to the shifter. you will probabally locate the subject someplace in there. in case you're nonetheless lined below the bumper to bumper guarantee then it would be a unfastened restore. verify you orignal place of work work.

BTSI solenoid to lock out shifter unless the brake is applied

hmmmmmm, thats a toughie, inside the car, or outside in engine compartment? If inside, it could be your E Brake handle. If outside, could be linkage to your A/T

Most likely it's the reverse lock-out. You didn't say what kind of car.

its called a Car!

if it shifts on the floor that would be a transmition if it shifts on the cullum that would be you and that is just a matter of oppenion