During Shadow's story, you can hear the voice actor for the G.U.N. soldier flub his line and repeat it again... and it's all unedited. Also, another G.U.N. soldier over Rouge's radio refers to the Scepter of Darkness as the "Book of Darkness" at one point, though the subtitles are correct.
The very first quest in the game is called "The winder of a shoemaker". It's definitely a typo, and was probably supposed to be "wonder".
Christmas Rushed: By far the most infamous instance of this, although information of its development made it clear it had the deck stacked against it—Sonic Team was forced to develop the game with half of its usual manpower—Yuji Naka had just left Sega and the other half of the team was busy working on Sonic and the Secret Rings, which was made to make up for the fact that Sonic 2006 was impossible to port to the Wii, and they had to make the game for two advanced, brand new consoles. On top of that, Sega wanted the game out in time for Sonic's 15th Anniversary, so Sonic Team was forced to rush the game out the door and disregard bug testing and quality assurance. The final result was a game that left a black mark on the series and Sega's reputation forever, and one of the most infamous video games ever made.
Creator Killer: A handful of Sega employees, including two of the game's three story writers, were fired from the company after this. In addition, Shun Nakamura, who was one of the leaders of the project and one of those story writers, became less involved with the Sonic franchise as a whole, and the controversy did not do Yojiro Ogawa and Yuji Naka (the latter of whom had departed Sega by this point) any favors for their careers; Takashi Iizuka was eventually put in charge of Sonic Team at the end of the decade to try and fix their mess.
A variation: when they reconvened to do Sonic Unleashed, Pokecapn's band usually referred to this game as "that other game" whenever the subject came up. Not surprising considering the effect "that other game" had on their sanity...
"NO! Shut up! Shut the fuck up right now! We vowed never to speak of that whole ordeal..."
They also refer to the magma-lizard enemies as "Dodongos" or "Thriller lizards" (after the funny dance they do when Silver telekinetically picks them up) and the flying magma enemies are "Fire Keese", after the flaming bat enemies in the Legend of Zelda, in keeping with Medibot's need to make everything fit into some other game's logic.
They dub the Demonic Spider black hole things from End Of The World "Time Dimensions".
They also give Sonic the name Rockman Classic/Rockman Blue, Shadow the name Rockman Charcoal, and Silver the name Rockman Steel / Rockman Chrome.
Mephiles is referred to as Mephistophiles, Satan, or Mephisto.
Referring to Silver's Psychokinesis as "Physics". As a result of their Sanity Slippage they use the word "physics" as a verb.
"My rings! They're getting physics'd!"
Keep Circulating the Tapes: While used copies aren't too difficult to find, Sega pulled all new copies from store shelves in 2010, so don't expect a rerelease in any form.
Manual Misprint: Perhaps unsurprisingly, considering the game was released as an Obvious Beta. One example, though — the manual states that the energy meter on the HUD is utilized in conjunction with the special gems that Sonic can buy at the shop in Soleanna, and that it is meant to drain with each usage of the gems. This is not true in the actual game; again, since it was released in an unfinished state, the energy meter never drains and as such Sonic is able to use the gems as much as he wants, when it is quite clear that it was never intended to be this way due to their abilities to break the game on unlimited uses.
Medal of Dishonor: Meta-example; the game won the "Worst Game of the Decade" award over Big Rigs: Over the Road Racing, and while X-Play had ripped apart a few Sonic games prior and made their disdain for the franchise at the time known previously, this was the first Sonic game to get one of their "Golden Mullets", their version of the Golden Raspberry Award that was taken from Aquaman (the first game on the show to get a 1 out of 5 score).
Milestone Celebration: It was intended as a celebration of 15 years of Sonic the Hedgehog. One of the rare cases where such a thing went terribly wrong, due to the game being rushed to ensure it was finished in time.
ZigZagged. SEGA fired the game's writers, apologized on its multiple issues, and later pulled it from stores entirely. All as an effort to forget that the game existed, and to hope that this problem never happens within the series again. However they weren't quite ashamed enough to write it out of Sonic canon forever, as several elements from this game appeared in later installments. Silver appears in a few later games like Sonic Rivals and one of the levels with a boss fight are in Sonic Generations, albeit with a major gameplay overhaul.
During the 25th anniversary event, all the Sonic games up to that point were listed and each game was given a cheerful description. When the list got to Sonic '06 its description was just the game's loading screening.
Sonic makes reference to the game in LEGO Dimensions, and makes it clear he'd rather forget about it.
Sonic:Lumpy Space Princess, huh? Last time I met a princess, Ió We... Y'know, I d— I don't quite remember. Never mind.
Early trailers and demonstration videos promised things like a day/night system, multiplayer minigames, and a "free mode" that allowed you to play as any character without storyline constraints. Several functions described in the manual are not actually in the game, including things like the implementation of the Action gauge (it is completely pointless for Sonic and is rather barebones for Shadow and Silver), "maturity" levels for Sonic's custom actions (there are none), and a flight gauge for Omega's hover (he can hover forever, there is no gauge). Most depressingly, code was found for a purchaseable gem that allowed Sonic to relive his glory days of going Super Sonic in-level, rather than just for the final boss. Such a thing was never used until Sonic Colors.
There were originally plans to create a Wii port of this game, but hardware limitations had them make Sonic and the Secret Rings instead. Ironically, SEGA taking half of '06's team to work on Secret Rings was a major contributing factor to '06's unfinished state.
A more minor standpoint, but it was implied in some unused quotes in the game that Sonic/Shadow gets sucked into the tornado in Crisis City and the other one has to rescue them, the two nearly thanking each other. In other words, it was implied they were going to be playable in each other's action stages.