What is the origin of the one-horned rabbit so common in fantasy and isekai stories?
It's possibly a reference to Bunicorn, an early-stage recurring enemy in Dragon Quest series.


Bunicorns are rabbits with a single horn, parodying the fabled unicorns. They have gray fur, and the horn is often a yellowish brown shade. [...].
As enemies typically encountered at the start of a game, Bunicorns are not particularly strong, though they typically outweigh the likes of Slimes [...].

The creature first appeared on Dragon Quest III in 1988 (Japanese NES release).
Considering that Dragon Quest is really popular in Japan, I think this is the most possible reason for using it as a beginner's monster, as an alternative to Slime.