This page explains in detail the interactions between the Famicom Robot series and the Tetris series.
Tetris DS[]
March 20, 2006
Famicom Robot
Tetris
One of Dr.Hector's instructions.
Dr. Hector from the Famicom Robot series often appears in the top screen of the Nintendo DS to give instructions and tips to the player. This is considered a type 1 link because he's physically in the game, since he talks about it, he's not just an image.
(This may be considered just a type 2 link, because he talks about Tetris as just a game, but it wouldn't work: to consider a fictional series within the fictional series, we should have seen a Tetris game within Gyromite or Stack-Up, instead we have Dr. Hector in Tetris. So, it doesn't matter if he's aware that Tetris DS is a game: he's in it, it's not the other way around.)
License[]
Tetris DS was produced by Nintendo, so they could freely include characters from their series in it.