This page explains in detail the interactions between the Metroid series and the Tetris series.
Tetris (NES)[]
November 1989
Metroid
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 Samus who's seen playing cello. The other characters are Peach, Bowser and the Mario brothers from the Mario series, Donkey Kong, Pit from Kid Icarus, and Link from Zelda (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
Metroid
Tetris
Tetris DS features six different modes, each one with themes from a different Nintendo game, and Catch Mode is the Metroid-themed mode (For details about other modes see Nintendo X Tetris#Tetris DS). It features a gameplay "opposite" to the classic Tetris, since the player moves and rotates the playfield to "catch" the tetrominos instead of the other way around. Sometimes Metroids fall with the tetrominos and cause the player to lose HP if they hit they playfield. During gameplay, genric images of Samus from the original Metroid fighting enemies from the game, scroll in the background.
This is the only game mode in the game that's considered to create a Type 1 link with the series since while the other modes only have elements from Nintendo games in the backgrounds, in Push Mode Metroids directly interact with the Tetrominos.
Additionally one of the themes past Level 10 in Standard Mode is based on Metroid, and precisely Level 12, with the top screen showing gameplay from the game, while the background of the playfield shows the sprite of Mother Brain and on the right of the playfield is a sprite of Samus.
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.