This page explains in detail the interactions between the Tetris series and the Zelda series.
Tetris (NES)[]
November 1989
Tetris
Zelda
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, including Link who's seen playing a flute, likely a reference to the flute item from the games. The other characters are Peach, Bowser and the Mario brothers from the Mario series, Donkey Kong, Pit from Kid Icarus, and Samus from Metroid (see Nintendo X Tetris#Tetris (NES) for other details).
This is considered a type 1 link because Samus appears phisically in the game's world, and not just as an image, even if he doesn't directly interact with the game.
Tetris DS[]
March 20, 2006
Tetris
Zelda
Link fighting Tetrominos in the game's title screen.
Tetris DS features six different modes, each one with themes from a different Nintendo game, and Mission Mode is the Zelda-themed mode (For details about other modes see Nintendo X Tetris#Tetris DS). It features standard Tetris gameplay, but asks the player to accomplish specific tasks, such as clearing a specific number of lines, or using specific tetrominos. The bottom screen shows the playfield, while the top screen shows a scroll with the mission and an example screen. The only Zelda elements are in the background, and specifically the top screen shows the background of The Legend of Zelda title screen, while the background of the playfield shows the entrance to the Moon dungeon, and on the right of the playfield is a rotating sprite of Link.
Additionally one of the themes past Level 10 in Standard Mode is based on The Legend of Zelda, and precisely Level 11, with the top screen showing gameplay from the game, while the background of the playfield shows the first oldman in the game with the Wooden Sword in front of him, and the same sprite of Link from Mission Mode is on the right of the playfield is a sprite of Samus.
Lastly, Link makes an appearance in the title screen, where he's seen fighting Tetrominos as if they were monsters. Only this little reference is consdered a Type 1 link.
Tetris 99 - Version 2.0[]
September 5, 2019
Tetris
Zelda
Update 2.0 of Tetris 99 added new background themes based on Nintendo games, each purchasable for 15 tickets in-game, and they include one based on The Legend of Zelda.
For details on all the featured themes see Nintendo X Tetris#Tetris 99.
License[]
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.
Similarily Tetris 99 was published by Nintendo exclusively for the Nintendo Switch, so they gave authorization to use Nintendo themes in the game.