I am sorry for starting to kung fu panda thing in my own thread. Lesson learned. Let's move on

The type of attack is listed in the description of the creature in question. You'll see for most beasts it is listed as a "melee weapon attack". An unarmed strike is also a "melee weapon attack".
One point that just occurred to me: The monk ability to get the bonus-action unarmed attack only happens when you take the Attack action on your turn. The bear's claw/bite routine is a special ability, Multiattack; not the same thing as taking the Attack action. So you can use the monk ability to get two claw attacks, or you can use Multiattack to get a claw/bite, but you can't stack them to get claw/claw/bite.
One point that just occurred to me: The monk ability to get the bonus-action unarmed attack only happens when you take the Attack action on your turn. The bear's claw/bite routine is a special ability, Multiattack; not the same thing as taking the Attack action. So you can use the monk ability to get two claw attacks, or you can use Multiattack to get a claw/bite, but you can't stack them to get claw/claw/bite.
A martial arts bear. Seriously?
D&D is not a universal fantasy kitchen sink.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.