In Tokyo, bicycles must be registered with the local authority. Registration costs just ¥500 (about $7) and is done once when you purchase a bike (new or used). When registered, a sticker is applied to the bike frame which is useful if the bike is lost or stolen. Bearing in mind bicycles use the same roads as cars or bike paths specifically marked out for them, registering a bicycle makes user-pays sense and if it is done for a small fee when a bike is purchased there would only be a small administrative impost on the government.
Japan appears to have a common law compensation or damages system and a recent case involving a brain injured pedestrian who sued the cyclist who caused the accident achieved notoriety when the pedestrian obtained judgment in the sum of over $1 million and enforced the judgment bankrupting the defendant in the process.
The Bicycle Safety Promotion Association has worked with underwriters to offer cycling insurance in Japan at a cost of ¥500 yen per month ($7) providing personal liability insurance (of up to ¥100,000,000 or $1,355,000), personal injury cover to the owner of the bike and a form of roadside assistance for repairs or punctures.