The PC version's limit does not affect the monster version (ability assumptions like that can fall under DM Fiat, but that's not valuable information if you're looking for an informed ruling).
And the Bugbear text block does not specify a limit - so there is none.
It would trigger once per hit.
However, Bugbears do not have Multiattack - they can only make one attack per round. Thus, "per hit", "per surprised creature", and "per battle" are functionally equivalent - creatures are never Surprised for more than a single round at the start of the fight.
A Bugbear Chieftain or other monster with both Surprise Attack and Multiattack would be able to trigger it as many times as it hit attacks on Surprised creatures.