The Fury Shadows sure do act a lot like Shadow Mario, running around and forcing Mario to cause enough damage so they drop the shine they're holding. Considering Fury Bowser and his powers are a result of Junior using the Magic Paintbrush and accidentally corrupting him with black goop, it's likely that's intentional.
Bowser Jr's paintings in the credits sequence reveal that Fury Bowser was unleashed after Jr doodled on Bowser while he was asleep and seemingly made him so unfathomably angry he became akin to a natural disaster. However, even Bowser doesn't seem to have that bad a temper when it comes to his son's antics, at most taking him off to the side to scold him. Considering Jr. was using the Magic Paintbrush he tricked E. Gadd into building for him, it's likely that the magic paint did something to Bowser's brain to cause his mindless rampage. If Mario hadn't knocked the goop off him, who knows what it'd have done to him if it was allowed to dry?
