It activates only once

The trait reads (emphasis mine):

When you damage a creature with an attack or a spell [...]

Notice how "attack" is written in lowercase. This indicates a single attack. Not an Attack Action. When you have extra attacks, you attack multiple times with one Attack Action. But when you activate Fury of the Small for example on your first attack of your multiple-attack Attack-Action:

you can't use it again until you finish a short or long rest.

This means you can only use this effect on one of your attacks. The same applies to spells.

Except when you use AoE spells

In the comments Erik raised a very interesting point for this discussion:

Can you elaborate the difference (if any) between a multi-attack spell like Eldritch Blast vs an area spell like Fireball? (The former makes multiple attack rolls, the latter only has one damage roll which is used against every target) ?

The difference is that an area spell is still one single spell and that you don't make a spell attack. The trait does not limit the effect to spell attacks, but instead to spells that cause damage. Instead the targets make saving throws. That means that if you use this trait with an area spell like Fireball you should be able to add the extra damage to every single target that takes damage from your spell. Note that some AoE spells also have a clause that mentions a target taking damage even if they make their saving throws. These targets would still be valid targets for the additional damage.

If you instead use the Eldritch Blast you have the same situation as I already described in the case of multiple-attack Attack-Actions: you attack multiple times with a spell attack. This means you can only add the damage to one of these attacks.

Personal Addendum

I feel like the described AoE damage increase is not really the intended use of the trait, but according to the Rules as Written you would be able to add the extra damage to any target that takes damage because of your AoE spell. As a DM I might rule that the trait should read:

When you damage a creature with a [weapon] attack or a spell [attack] [...]

This would disallow AoE spells and other spells that use saving throws. It would only apply to real attacks against single targets and could only be used once.

Another way of changing the trait might be:

When you damage a [single] creature with an attack or a spell [...]

This would make it clear that the damage has to be focused on a single creature. In general this would disallow AoE spells, but then again it may be the case that it just so happens that only one enemy was caught in your Fireball, which would allow this again. While the second version might be interesting for some people as a way of changing the trait to deal with the problem of applying the damage to multiple enemies, I would recommend to use the first "fix" to simply disallow AoE spells.

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.