LEDs can be strange. The green ones can even produces a small voltage. Saw someone using them as a light sensor for a solar tracker.

The whole point is capacitance, not sure how the gizmo you are talking about works but I will bet the plastic is the di-electric and your finger is the positive side of the capacitor.??

Have you ever done the Leyden Jar experiment? Build a capacitor, charge it, take it apart with out discharging, completely separate the components, put it back together and it will still have a charge.