This page explains in detail the interactions between the Donkey Kong series and the Tetris series.
Tetris (NES)[]
November 1989
Donkey Kong
Tetris
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 Donkey Kong who's seen playing bass drum. The other characters are Peach, Bowser and the Mario brothers from the Mario series, Pit from Kid Icarus, Samus from Metroid and Link from Zelda (see Nintendo X Tetris#Tetris (NES) for other details).
This is considered a type 1 link because Donkey Kong 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
Donkey Kong
Tetris
Donkey Kong throwing Tetrominos at Mario in the game's title screen.
Tetris DS features six different game modes, each one with themes from a different Nintendo game, and Push Mode is the Donkey Kong-themed mode (For details about other modes see Nintendo X Tetris#Tetris DS). It features competitive gameplay, with the player and the CPU stacking blocks on the same field, but respectively from the top and the bottom, trying to "push" the stacked blocks toward the opponent. During this mode theme from the 25m level of Donkey Kong plays and the background shows a higher version of the level with Mario walking around and jumping over barrels. On the left items from the games can be seen: two Hammers, Pauline's purse and her hat.
Additionally one of the themes past Level 10 in Standard Mode is based on Donkey Kong, and precisely Level 13, with the top screen showing gameplay from the 25m level, while the background of the playfield shows a still image from level 75m and the sprite on the right of the playfield is Pauline.
Lastly, Donkey Kong and Mario make an appearance in the title screen, where the ape throws Tetrominos at Mario and he smashes them with the hammer. Only this little reference is consdered a Type 1 link.
Tetris 99 - Version 2.0[]
September 5, 2019
Donkey Kong
Tetris
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 Donkey Kong.
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.