I just tested all weapons on a Jumbig and found that associate weapons can still deal different amounts of damage. On the Jumbig, Toad does 4 while Gravity does 1 and Centaur does 2. Plant does 2 while Star and Skull do 1 each. Napalm can one-shot the Jumbig while Drill and Knight need two shots to take it down since that enemy is weak to Drill and its equivalents.
The associate system may explain the 2 damage regarding most of MM6's weapons, but the above example suggests that associates don't need to deal the same damage.
Also, Ring can pierce shields, but not Gyro or Tomahawk, which could mean associates don't really need to share the same properties either.