The white Donkey Kong puppet fighter represents the Polar Bear, an enemy in Ice Climber.
The stage and music represents the Summit taking place on top of an icy mountain.
The Freezie item likely references to the sliding ice that repairs the gaps made in the platforms. The first and third battle conditions and the puppet fighter's frequent use of Instadrop references to the Polar Bears being powerful enemies whose stomps causes the screen to move upwards.
In World of Light, the Polar Bear spirit must be defeated to access the Ice Climbers fighter, which references both the Polar Bear and the Ice Climbers appearing in the titular game.