There's a thing called "Magitek," which means that technology is powered by magic. It makes sense of having a magical horse-bike to exist here, due to the game takes place (chronologically in my opinion, despite not playing this game) IN DA FUTURE! I mean, what do you expect when there's lazer-spewing "octobots" roaming in the land of Hyrule, and the Master sword is rusting.

Considering all the psuedo-magic technology we see throughout the game, a simple bike isn't that far fetched.