Are the beast shape weapon attacks of a Druid, simple or finesse
... especially in conjunction with Monk multiclassing
The weapon attacks of a creature such as a Beast are 'Natural Weapons' and each entry for those attacks found under the Attack heading for each beast stat block tells you what kind of attack it is.
For instance, a Claw and Bite attack count as a 'Melee Weapon Attack'. Nowhere in the description for these natural weapon attacks do they count as either Finesse OR Simple natural weapons. Those are weapon types attached only to weapons found on the weapons table of the PHB.
Some attacks, such as a spike or spit attack count as a Ranged Weapon Attack for purposes of abilities or skills.
In conjuction with Monk multiclassing? No weapon attack (the natural attack options each beast shape allows you) you could ever make in a Wild Shape form would count as Finesse or Light or Monk Weapons, but they do count as a weapon attack.
Alternatively a 'Unarmed Strike' can be a headbutt, punch, kick, or other forceful blow. I think a bear is capable of a forceful blow... don't you? If only from a headbutt or body check. Which would then use your Monk's Martial Arts damage die and any attached abilities if the form is physically capable of the feature.