The name is the most important part, as it gives them a way to look for games that have their name but aren't theirs. But you're mostly right that they actually go after any game that has the same gameplay and is relatively popular, but only when they can easily issue a DMCA takedown. It's not like they ever said that this was their policy but you can see it bear out time and again.

TETR.IO and Jstris are insanely popular games that have been up for years with no action against them since they aren't hosted on another platform/marketplace.

This is also why Apotris got 2 DMCA takedowns on both Github and itch.io 3 months apart from each other, but after hosting the game on my own site, they haven't taken any further action in the past 2 years while it has continued to grow in popularity.

The shapes are called tetrominos, four squares connected orthogonally. Tetris the game, was created with pentominoes (5 squares) and scaled down to tetrominos. The game was inspired by physical puzzles where fill a rectangle with the shapes. So these shapes existed before Tetris.

Would they go after the shapes? Possibly. See "Tetris Holding, LLC v. Xio Interactive, Inc.".

Which is crazy, because the shapes are just an idea, and you can't copyright ideas. They also couldn't be a trademark infringement (IANAL, though) if you don't use them in your branding.

I could imagine the specific combination where the shapes colors match the original could get into trademark territory, as the colors have no in-game requirement to be those specific values.