Super Mario: Im Rausch der Geschwindigkeit
In the Club Nintendo comic Super Mario: Im Rausch der Geschwindigkeit, Mario is seen passing a vehicle apparently driven by Captain Falcon. While the vehicle doesn't perfectly corresponds to the Blue Falcon, since this is an official Nintendo comic, and the difference is probably just an inaccuracy, this can still be considered an in-universe link.
Super Mario-Kun Volume 4
November 28, 1992
In Volume 4 of the CoroCoro Comic manga Super Mario-kun, Mario finds himself in Hyrule, the world from the Zelda series, but at the end, after finally escaping, instead of returning in his own world, he ends up in the world from F-Zero.
Super Mario RPG
March 09, 1996
In Super Mario RPG, toy models of the Blue Falcon and the Fire Stingray vehicles from F-Zero can be spotted on a shelf in the back of Hinopio's shop.
July 14, 1998
The character Mr.EAD, first introduced in F-Zero X, has a moustache and wears a Starman belt. These are said to be references to the Mario series.
Also, he's an android created by "the mysterious development group EAD", hence his name, this is a reference to Nintendo Enertainment Analysis and Development (Nintendo EAD), the studio that created F-Zero and Mario games.
The first Joker Cup course is called Rainbow Road, and it is, in fact, the same exact Rainbow Road from Mario Kart 64. The only differences between the two are due to the different gameplay.
July 25, 2003
Among the various "emblems" players can use to customize their vehicles are images of the Super Mushroom, the 1-Up Mushroom and Mario himself.
Mario Kart Wii
April 10, 2008
A small kart called "Blue Falcon" and looking very similar to the F-Zero vehicle of the same name is unlockable in Mario Kart Wii. Its colors are the same of the original Blue Falcon when Baby Mario is the pilot.
Mario Kart 8
November 13, 2014 (Ver 3.0 Update); December 6, 2014 (Captain Falcon amiibo)
Update to version 3.0 of Mario Kart 8 added amiibo support to the game: scanning an amiibo from the Super Smash Bros. series unlocks Mii suits based on the scanned character. This includes Captain Falcon.
For info on other amiibo outfits see here.
The update also set the game up for the The Legend of Zelda × Mario Kart 8 DLC pack, which includes some F-Zero content:
The Legend of Zelda × Mario Kart 8
November 13, 2014
One of the new downloadable tracks is Mute City, based on the iconic location from the F-Zero series, most closely resembling the Mute City - Twist Road track from F-Zero GX.
Referencing the gameplay of the original game, the track is entirely in anti-gravity and there are Dash Pads and Recharge Pathways, which give coins in this game; the count downat the beginning is also based on the F-Zero countdown. Signs and buildings throughout the course depict and reference the various F-Zero racers, including Captain Falcon, Samurai Goroh, Dr. Stewart, Pico, Jody Summer, and Mr. EAD.
One of the new karts in the game is a new version of the Blue Falcon kart, returning from Mario Kart Wii.
Animal Crossing × Mario Kart 8
April 23, 2015
The third DLC pack for Mario Kart 8 added a second track based on a recurring F-Zero location, Big Blue, most closely resembling the Big Blue - Ordeal track from F-Zero GX.
Like Mute City it's entirely in anti-gravity, it includes Dash Panels and Recharge Pathways. Additionally, upon reaching the final lap, the voice "Yeah!! The final lap!" from F-Zero X will be heard.
Super Mario Maker
September 10, 2015
One of the unlockable costumes that Mario can wear in the game by using a Mystery Mushroom is based on Captain Falcon, and it's unlocked in the 100 Mario Challenge mode or alternatively by using the Captain Falcon amiibo from the Super Smash Bros. series. It's an 8-bit design based on Falcon's recurring appearance, when worn it replaces the sound effects with sounds from F-Zero.
All rights for both series are held by the same company (Nintendo).