The first official crossover between a Nintendo property and a Sega property is often assumed to be either Mario & Sonic at the Olympic Games or Super Smash Bros. Brawl. While these are indeed the first Nintendo-Sega crossovers within the gaming medium, there have been instances prior of crossovers in supplementary marketing material:

• A 1994 advert made by DIC for animation industry insiders showing their global reach through the Panda Club and Dragon Club TV blocks in China shows Sonic The Hedgehog waving through a TV screen while Mario pats him on the back, joined by Madeline from the Madeline book series and Billy Lee as seen in the cartoon adaptation of Double Dragon.
• The infamous Fight for the FoxBox promotional film made by 4Kids Entertainment in 2003, which utilizes existing anime and cartoon clips to make a new crossover story, shows Dr. Eggman and King Dedede as allies alongside Dial-bolic from Ultimate MUSCLE, Dr. K from Cubix: Robots for Everyone, and Shredder from Teenage Mutant Ninja Turtles working for Wayne from The Cramp Twins in an attempt to shut down the titular television channel. Although the villains do interact in this half-hour film, the heroes mostly do not, with Kirby and Sonic never sharing screen time in particular.

This is, of course, discounting instances of unauthorised parodies and mockery at the height of the console wars.