I would say that would be the pedal stop rubber, which also presses on the switch. The brake pedal in the untouched position sits against it keeping the lights off. With this gone, there is nothing for the switch to rest against, causing the lights to illuminate. The cruise control thinks that the brakes are applied, which explains why it will not work.
This should be the genuine part number for it: CUSHION(FOR STOP LAMP SWITCH) 90541-06036.