These seals, always fastened at the top-left of the rear license plate of full-sized vehicles* (普通自動車) registered in Japan, are anti-theft measures, and they're protected by law. Removing one without explicit permission (such as changing the plates on a car, having a car sent out of country, or having a car destroyed) is illegal. However, if you relocate to a different prefecture of Japan, it is your duty to have your plate changed at the appropriate office.
*(In other words, they aren't issued for kei cars.)
By the way, when screws for a new license plate are issued, the seal isn't handed to the owner of the vehicle. In standard procedure, your shakensho (vehicle registration document) is kept in custody of the Land Transport Bureau until you've screwed on the plates yourself. Then, they'll come and check your car, looking at vehicle identification numbers printed under the hood and on the body, and when they're finally satisfied that it's really the car identified in the shakensho, they'll pop the seal into place themselves.