Nope, BS.

The principals you're thinking of are Faraday's Law and Lenz's Law. An inductor stores energy (current) in its magnetic field. As the current is used, the magnetic field collapses.

Without getting into an overly long and boring technical discussion, an inductor by itself cannot output more energy than you put in. In fact, (due to parasitic capacitance, hysteresis and wire resistance), an inductor can't even put out the same amount of energy as you put in. However, if you influence the inductor with a magnetic field, then it can convert the magnetic field to electricity (as happens in a generator).

The small amount of energy a coil can store would not be sufficient to even light-up a 30W light bulb. It might be able to power a low-voltage neon lamp for a second or two. But not the setup shown. The coil is too small, too loose and uses thick, stranded conductors that would have very high parasitic capacitance and hysteresis.

What is interesting if they can be done effectively, is Piezoelectric or Pyroelectric energy harvesting. This is generating small amounts of current from normal movement or temperature variations and storing it over time in a capacitor or battery, then using that to power electronic devices. Kinetic watches are practical examples of piezo energy harvesting.

So far, Newton's Third Law and the law of conservation of energy have not been overturned. The video is a fraud.