Broken Base: The whole "two drivers at one time" mechanic. Either it's one of the most innovative features to be introduced to the entire Mario Kart franchise and needs to make a return, or it's one of the worst aspects about Double Dash and it's a good thing it has yet to be brought back. There are also those who were completely indifferent to the whole mechanic, but there have been some debates among fans of the first two camps on if the feature should or should not return in a future entry.
The lightweight characters are by far the most used due to being out-and-out Game Breakers with karts that have very high acceleration and handling, and are tough to hit with items. The crown however goes to Baby Mario and Baby Luigi, as they also have a devastating Special Item in the form of the Chain Chomp.
Koopa and Paratroopa's special items are triple red and green shells, which are normally items anyone can get, but in this game they have exclusive access to them. The Chain Chomp might be powerful, but it doesn't come close to the Boring, but Practical "more things to throw at the enemy/hold behind you to protect yourself".
Franchise Original Sin: in Mario Kart 64 and its GBA successor, Super Circuit, the Spiny Blue Shell worked like a red shell, but it hit every driver in front of the one who threw it and finished with the one in the first place; this game, however, marks the debut of the (much more hated) flying version of the item, that only hits the first-place driver with an explosion that takes five to ten seconds to recover. By the time the first-place driver is back on track, the second-place driver is likely to have already surpassed him, and usually said second racer is the only one taking any advantage of the weapon. It took Nintendo two more games (Mario Kart DS and Mario Kart Wii) to either realize the mistake themselves or to notice the item's infamy among fans before reverting the item to its non-flying form (while still exploding when hitting the head racer) in Mario Kart 7.
Toadette, since the only precedent of a Mario character in a series apart from the platformer games was Waluigi. Also, this was her first appearance.
Also Paratroopa, who is a partner for Koopa.
Petey Piranha and King Boo are among the first boss characters since Bowser in Super Mario Kart to be playable, so no one saw them coming (even though Birdo, Diddy Kong, and Bowser Jr. debuted in this game as well, for many players the debut was more plausible).