Technically registering your bike is mandatory, and all sold in Japan should be registered (for instance when you buy a bike here typically you do a bit of paperwork at the shop and get it registered at the time of sale). You get a little yellow sticker which shows your bike is registered. To my knowledge there is no penalty for not having your bike registered other than the police being able to hassle you ('is this bike yours?' 'it's not registered!' 'where did you buy it?' etc.)
I've never been stopped or hassled but some folks have (I've never been stopped of bothered by the police at all, but I started coming to Japan in my early 30s ... no one really bothers me). Given that registration is about 500 yen (or 5 USD or so) I want to register so as to not ever have any hassle. A minor inconvenience to potentially avoid future inconveniences.
Just because something is not normal in the US does not mean it is not common elsewhere, US norms are just that, norms in the US.