Thanks for the nice comments :)
Update 03
So I haven't had that much time to work on the game jam this weekend, but I realised what I was doing wrong. I forgot that GameMaker distinguishes between objects and specific instances of those objects. My tetrominoes are supposed to be arrays of four blocks each, but instead they were arrays of the concept of "block" four times. So instead of moving a tetromino, the above gif collapses all the blocks into the first instance of "block" it can find and moves that around.
I managed to fix it, and now it works better:
Still a lot of things I haven't coded yet - you can stack the blocks on top of each other, you can pick a block up in the same step that you put one down, blocks still need to snap to a grid - but I'm happy so far. Luckily, GameMaker just put out a tutorial for only clicking on the topmost instance in a stack of objects, which I hope will help!