There‘s also an excellent Gameboy Color romhack of Dr Mario that supports multiplayer. Recently tried that out with my girlfriend and it was a lot of fun.
Edit: this is it for anyone interested. Looks like even the original version for the Gameboy supports multiplayer.
When you get lines, your opponent's stack pushes a line with a gap up from below, except when you get a Tetris, which pushes four lines (with the gap aligned, so you could Tetris back and forth).
You had an indicator for the max height of your opponent's stack next to yours.
If you haven't experienced multiplayer tetris from the modern remakes you may be in for a treat. It's a lot of fun when you're up against someone of similar skill. At least one switch version includes ranked matchmaking (Puyopuyo Tetris).