Here you can speculate about the Target Test modes.
Let's remember that character specific Target Tests themes often reflect several elements in the character's own games (Mario's has pipes, DK's has a construction theme on it, Link's is based on the Temple stage, etc) so I'm gonna speculate the setting of some character's Target Tests:
Kirby: Based on the HUD room in Kirby & the Amazing Mirror, refrencing the original Super Smash Flash.
Meta Knight: Based on the insides of the Battleship Halberd, Meta Knight's ship.
Megaman: Based on the insides of Wily's Castle, with moving and dissapearing platforms.
Sonic: Based on the Bonus rooms in the first Sonic game, with Bumpers and one Target traped inside a jail made of gems (refrencing the fact that the Chaos Emeraldswere inside these little "Gem jails")
Goku: Based on the King Kai's planetoid from Dragon Ball Z (a.k.a. the place were Goku went to train in the Saiyan saga)
Bowser: Based on the Bowser's Castle stage sans the Thwomps and the Lava.
In Melee, there was a trick for alternate music in the Target Test, so the music doesn't feel that repetitive.
If that feature is included, then this is my speculation for the music:
-Mario: Super Mario Bros 3 (Melee) or Mushroom Kingdom III (SSF2) if only game's soundtrack counts.
-Peach: Rainbow Cruise (Melee) or Peach's Castle (Melee) for same reason as above.
-Bowser: Airship Fortress (SMB3) or Bowser's Castle (SSF2) for same reason as above.
-Yoshi: Yoshi's Story (Melee)
-Donkey Kong: Gangplank Galleon (SSF2) (Skip to Bonus part of the song)
-Wario: Wario Ware inc Medley (Brawl) or Emerald Cave (SSF2) for same reasons as above
-Marth: Fire Emblem (Melee) or Fire Emblem (Brawl) for same reasons as above
-Link: Great Bay (Melee) or Skyward Voyage (SSF2) for same reasons as above
-Zelda/Sheik: Temple (Melee)
-Lloyd: Final Destination (Tales of Symphonia) or Beat the Angel (ToS) for same reasons as above
-Kirby: Mirror Chamber (SSF2)
-Meta Knight: Meta Knight's Revenge (Brawl) or Mirror Chamber (SSF2) for same reasons as above.
-Chibi Robo: Desk (SSF2)
-Isaac: Golden Sun (Brawl)
-Samus: Brinstar (Melee) or Crateria (SSF2) for same reasons as above
-Zero Suit Samus: Vs Ridley (Brawl) or Phase 8 (SSF2) for same reasons as above
-Megaman: Castle Wily (SSF2 0.6) or Skull Fortress (SSF2) for same reasons as above
-Bomberman: Bomb Factory (SSF2)
-Sora: Sinister Sundown (Kingdom Hearts)
-Sonic: City Escape (SA2) or Green Hill Zone for same reasons as above
-Tails: Casino Night Zone
-Fox: Corneria (Brawl) or Sector Z (SSB64)
-Captain Falcon: Mute City (Brawl) or Sand Ocean (SSF2)
-Ness: New Age Retro Hippie (Earthbound) or Hi Hi Hi (SSF2)
-Mr. Game & Watch: Flat Zone 2
-Black Mage: Chaos Shrine (SSF2)
-Pikachu: Battle Theme (Gold/Silver, Melee) or PKMN Colloseum (SSF2)
-Jigglypuff: Pokè Floats (Melee) or Silph Co (SSF2)
-Ichigo: Bleach Medley (SSF2)
-Naruto: Rising Fighting Spirit (Naruto) or Hidden Leaf Village (SSF2)
-Goku: DBZ Opening (Instrumental) or Namek (SSF2)