Tetris (Game Boy)
June 14, 1989
When playing a two-player linked game, Player 1 is rapresented by Mario and player two by Luigi and at the end of each round the winning player's character is seen exulting, while the losing player's character is seen crying. In this scene each player always sees their character in the foreground and the other player's in the background, and the winner player sees them facing the screen, while the loser sees them from their back, giving the illusion of the same scene being seen from two different perspectives.
Finishing the game in Type B mode, will show a cutscene of some people dancing and playing instruments. While they're generic characters in the Game Boy version, in the NES version they're replaced with Nintendo characters. These include Peach clapping, Bowser playing accordion, Mario jumping and Luigi dancing. The other characters are Donkey Kong, Pit from Kid Icarus, Samus from Metroid and Link from Zelda (see here for other details).
This is considered a type 1 link because the characters appear phisically in the game's world, and not just as images, even if they don't directly interact with the game.
3-in-1 Super Mario Bros./Tetris/Nintendo World Cup
This game cartdrige contains three NES games: Super Mario Bros., Tetris and Nintendo World Cup. All versions of the games are identical to the original, the only addition being obviously a game select sceen at the start.
Tetris & Dr. Mario
This game cartdrige containins enhanced versions of the NES games Tetris and Dr. Mario. The gameplay in both games remains the same, they only feature improved graphics. The game also includes a new "Mixed Match" mode where the players have to play Tetris mode B then Dr. Mario and finally Tetris mode A.
Mario Artist: Polygon Studio
August 29, 2000
The "Block" microgame in the game is based on Tetris, since it features a falling block that the player must direct toward an opening in the bottom, filling the lines.
march 20, 2006
Tetris DS features six different game modes, each one with themes from a different Nintendo game and Standard Mode is considered the Mario-themed mode (For details about other modes see here). Actually the last half of levels (11 to 20) feature themes from other Famicom/NES games, but due to the increasing difficulty and speed of gameplay, this mode do is played with Mario themes for the most part. Likely due to it being the basic mode, Mario and a Lakitu are also shown on the cover of the game. Additionally, the only thing that makes this a Type 1 link is Mario's appearance in the title screen, where he's seen fighting Tetrominos as if they were Goombas.
Standard Mode features the classic Tetris gameplay with the background showing images from Nintendo games. The play field is only on the bottom screen of the Nintendo DS, while the top screen shows images from the game, with redesigned backgrounds. Additionally the background of the playfield shows a still image from the game while to its side there's a rotating sprite from the game.
The theme shown changes in every level, that means every ten lines cleared, and the first ten levels are all themes from Super Mario Bros. and Super Mario Bros. 3, while each level from 11 to 20 shows images from a different Nintendo game (see here for details about them).
Details for the Mario ones, indicating top screen animation / playfield background / sprite at the right of the playfield:
The NES version of Tetris, as well as Tetris DS were produced by Nintendo, so they could freely include characters from their series in them and boudle them with other games. "Block" in Mario Artist: Polygon Studio is only vaguely inspired on Tetris, so Nintendo didn't need an authorization for it.