In early radio days, transmitters were literally generators spun really really fast. Even earlier, spark gap was used. High voltage between gaps, or inductively induced, like you are doing or some other ways. Coils store energy. When it is suddenly released, it comes out as high voltage.
As to legality, what he has is nothing more than a buzzer. Unless he connects antennas to it, it won't go too far outside of his room or house. Perfectly legal.
DO NOT connect it to spectrum analyzers or even oscilloscopes! You can connect a short piece of wire in such instruments and place it few meters away. There is enough energy saved in that coil to blow away the FET or whatever front end you might have.