So this is referring to trademark (the name Wordle) and copyright — but not patent. It makes sense to go after people who are using the same actual name, since this clearly infringes the trademark, and because if you do not enforce ("police") your mark against minor players, you can end up losing the ability to enforce it against major players.
But the copyright bit is a bit novel from my perspective (lawyer, but not copyright lawyer). If you had asked me what a copyright claim about Wordle would be about, I would have said the precise code. I might have wondered about the specific word lists, even though these would probably fail the "phone book" test (don't remember the case, but these were deemed uncopyrightable). I never would have thought about the tile layout and color scheme. That seems more like what I think of as "trade dress" [1] or perhaps something related to patent (which wouldn't apply here, unless the original Wordle owner had filed for patents a long time ago.
Are there any copyright lawyers who can elucidate how the tile layout and color scheme might be subject to copyright law? I assume the NYT has good lawyers, and has thought long and hard before going after folks on github...