Although the Game Over screen is the same for all minigames (with 2 exceptions, already covered in this video) the music is completely different.
A standard Game Over music is for Spring Breeze, Great Cave Offensive, Revenge of Metaknight and Dynablade.
But a different noted music of the Star Rod screen from Milky Way Wishes is played in that minigame.
And the 2 losses screens, from Gourmet Race and The Arena. The music is the same on these two minigames.
Update: Annotations Added just to make specific explanation of this video and the Game Overs of Kirby. lol.