Canon

Franchises and games are considered canon or non-canon based on how closely linked they are to "core" Mario games. In general, games that are clearly crossovers are not considered canon. Games that take place in the "real-world" are also not considered canon.

Mii
Mii is the flagship series of the Wii, and extends onto newer consoles as well. Wuhu Island is a location that exists in the Mario Universe, as seen in the Mario Kart 7 Star Cup and Flower Cup, however, the Mii series is considered large enough, divergent enough, and "meta" enough that its games are generally not considered canon to Mario. Miis are seen cheering for Mario in Mario Kart Wii and Mario Kart 7, and so are considered a group of humans from Wuhu Island.

Game & Watch
All but two of the games in the Game & Watch franchise has been re-released as Mario games in the Game & Watch Gallery series.

Excite
Excite, best known for including Excitebike, is a series of games that has some ties to the Mario franchise. The Excitebike arena appears in the Super Bell Subway map and as a Mario Kart 8 Deluxe stage, and the game Excitebike: Bun Bun Mario Battle Stadium is thought to take place there. As such, parts of the Excitebite series are canon to the Mario franchise. However, the Excite franchise as a whole is large and separate enough to be considered generally "non-canon", so games such as Excitebike 64, Excite Truck, and Excitebots: Trick Racing are not considered part of the universe.

R.O.B.
R.O.B. appears as a playable character in Mario Kart DS, and is referred to in the Wario games, meaning he exists as a concept in the Mario Universe. However, appearances outside of Mario games are not considered part of the Mario franchise.

Rhythm Heaven
Wario games have often referenced the Rhythm Heaven series, and so some Rhythm Heaven characters exist in the Mario Universe. However, the primary crossover game, Rhythm Heaven Megamix, is not considered canon due to the extensive crossover nature of the game, nor is the rest of the Rhythm Heaven series.

Super Smash Bros.
Super Smash Bros is explicitly a crossover series, and so is considered non-canon. Even if it weren't, the characters in Smash Bros are toys, as shown in the first game of the series.

Sonic the Hedgehog
Sonic is best known to join Mario in the Mario & Sonic at the Olympic Games series. This is clearly both a crossover series and a series that mostly takes place in the "real world", and so is not considered canon.

Rabbids
Mario joins with the Rabbids in Mario + Rabbids Kingdom Battle, however, this is certainly a crossover game. The Rabbids franchise is also large and highly divergent from Mario.

Dragon Quest
The Mario franchise crosses over with Dragon Quest in Itadaki Street DS, the Fortune Street series, and Mario Sports Mix. However, Dragon Quest has a large, complex, and highly divergent canon of its own, and so all of these games are considered crossovers rather than canon.

The Legend of Zelda
Link and Hyrule appears as a character and kart course, along with the Animal Crossing and F-Zero series, in Mario Kart 8 and Mario Kart 8 Deluxe, however, as part of crossover DLC, they are considered non-canon.

Animal Crossing
Similar to the Zelda franchise, Animal Crossing appears as characters and a kart course in Mario Kart 8 and Mario Kart 8 Deluxe, however, as part of crossover DLC, they are considered non-canon.

F-Zero
F-Zero joins Animal Crossing and Zelda with Mario in the Mario Kart 8 and Mario Kart 8 Deluxe. The series is considered non-canon as it is included as part of a crossover DLC.

Mario Edutainment
Games that are intended to teach "real-world" facts are not considered canon. Such games include "Mario's Time Machine" and "Mario is Missing!".

Final Fantasy
Final Fantasy briefly crosses over with Mario alongside Dragon Quest in Mario Sports Mix. For these reasons, the game and series are considered non-canon.

Punch-Out!!!
Mario seemingly stars as a referee in certain games in the Punch-Out!!! series, however, since the referee is never confirmed as Mario, and the series is highly divergent and takes place in the "real world", it is not considered a canon appearance.

Spyro
Mario has crossed over with the Spyro series in Skylanders: SuperChargers Racing. This is considered a crossover game from a separate franchise, and so is not canon.

Banjo-Kazooie
This series, along with Conker's Bad Fur Day, appear in the Diddy Kong Racing series. This has led to the game being considered non-canon, as the game is not only a crossover but also not a significant part of Donkey Kong canon.

Conker's Bad Fur Day
Similar to Banjo-Kazooie, this series appears in Diddy Kong Racing, leading to the game being considered non-canon.

SSX
Mario, Luigi, and Peach appear in SSX on Tour in a cameo appearance, but as this is considered a non-canon crossover.

NBA Street
Mario, Luigi, and Peach make an appearance in this "real life" basketball series, but besides taking place in the "real world", this is a crossover cameo in a separate franchise, and so is considered non-canon.

The Legendary Starfy
Wario makes an appearance in Densetsu no Starfy 3, but the series diverges enough from Mario that it is considered a cameo in a separate series.

Captain Rainbow
The game "Captain Rainbow" features Birdo prominently, however, this is considered a cameo appearance in a game not part of Mario canon.

Mario Kart Course Information
Many Mario Kart courses (and battle arenas) have been remade, and so may have adjusted canon. Included here is a list of all courses, with information on courses that have been remade. Not included in this table are 16 non-canon courses: Pac Mountain, Pac Labyrinth, Mute City, Hyrule Circuit, Animal Crossing, Big Blue, Urchin Underpass, New York Minute, Tokyo Blur, Paris Promenade, London Loop, New York Minute 2, Tokyo Blur 2, Paris Promenade 2, Vancouver Velocity, and London Loop 2.