"The Big Rumble" is likely a reference to Smash Brothers, known as Dairantou Smash Brothers in Japan (大乱闘スマッシュブラザーズ).
大 = big, 乱闘 = fight/fray
The Nintendo 64 console, SNES/Super Famicom controllers, and "Donky / Donkey Kong" are also clear Nintendo references.