That's still how they do it in Japan. Registration is mandatory for all bicycles. From what I hear you need to bring proof when there's a transfer of ownership, etc.
Even if you do get stopped for genuine wrong-doing, the worst that happens is you get a talking to or a citation at the nearest kōban. Better yet, if you learn even the tiniest bit of Japanese that will generally set the policeman at ease.
Now, some "gaijin" saw a business opportunity there and basically this level of safety is over in most big cities. I think it's totally understandable that they target foreigners. Japan is one of the safest places in the world and they want it to remain that way.
This has stopped recently though... I am wearing a cycling kit, switched to a road bicycle, and riding in traffic. So likely I've dropped out casual thief profile.