This page explains in detail the interactions between the Qix series and the Tetris series.
Qix (Game Boy version)
April 13, 1990
In multiplayer mode, Player 1 is represented by Mario and Player 2 by Luigi, the brothers from the Mario series. (see here for details) This is based on the multiplayer mode from the Game Boy version of Tetris, that also featured the Mario brothers. (see here for details)
Specifically in both games after each round the winning player's character is seen exulting, while the losing player's character is seen crying, but each player sees the scene from the opposite perpective, with their own character in the foreground.
The only difference is that in Qix the character's faces have been redesigned to better represent them and be more recognizeable.
License
Both the Game Boy version of Tetris and the Game Boy version of Qix were created by Nintendo.