I'll try to come up with something for all characters...
Mario: See above^
Bowser: Only character who can let the opponent control his movement
directly.
Again, the opponent can control Kirby, Dedede, and Diddy when they use their grab specials.
Diddy: Only character with an attack that gets canceled by any other attack with knockback (peanut).
Not quite sure what you mean here. If you outspace someone you can cancel any of their moves by one of yours...
Link: Only character with projectile with a windbox.
Sheik: Uuh
Ganondorf: Only character with three grab moves.
ROB: Only character that doesn't instantly regain his upB after landing.
King Dedede: Only character with a potentially infinite single hitbox (up B),
Given infinite downward space, Yoshi's and Bowser's Bombs wouldn't? Also Peaches open parasol?
Fox: Only character that gains height by using an aerial i mid-air (fair).
Falco: Only character who performs a neutral B faster in the air?
Wolf: ???
Falcon: Only character with a taunt that changes speed depending on the direction he's facing (dtaunt).
Pikachu: idk
Ike: Does Quick Draw have any frames of super armor? If so, Ike is the only character with only Super Armor/Invincibility Special Moves.
I don't believe it does
Sonic: Only character with a homing melee attack.
Kirby can use it as well though...