In the main menu when I select an option and the camera pans, it actually detects the correct buffer, presumably because the plane temporarily disappears.
However, as soon as the animation is done ReShade goes back to detecting this weird plane as the depth buffer.
I can't find any solution to this anywhere on the internet and I'm guessing nobody else here has run into this problem. It prevents me from using and kind of depth related shaders, including this cel-shading one in the game.
EDIT: I figured out how to fix this. You have to download the version of ReShade with full add-on support and then checkmark "Copy depth buffer before clear operations" in the add-ons tab.