We have come a long way from the Like-a-Bike.
Balance bikes are called runbikes in Japan, and so the competitive racing circuit is known as the Runbike Challenge Series. It is organized by the RFJ, Runbike Foundation Japan.
From the URL, RCS began in 2013. Kids around the country race in age classes from 2-5yo, working their way toward the All Japan Championships. Dads are heavily involved, but so are girls. Runbike racing really is a family affair.
There are videos on YouTube that rival muttonbustin' for their sheer wtf-ill-advised awesomeness. The related links will lead you down a runbike rabbit hole.
There also appears to be a branded competitor, the Strider Cup, sponsored by the company, which only allows owners of its product to enter. The trailer for the 2016 Osaka Round is the thrill of victory, the agony of defeat, and the lingering gender strait jackets that deploy a squad of 2yo girl cheerleaders for a 2yo boys' bike race. This whole thing feels like fun, but also like folly, and I cannot endorse it any more than I can look away.
Runbike Challenge Series [rcs2013.com]
The Strider Cup [strider.jp]
Oh look, the Strider Cup® is global. [striderbikes]
Random runbike racing video vortex entry point: RCS 2013 Grand Champion Heats 5yo Class [youtube thanks dt hero matt]