The point here is that the anticheat solution needs to be written for a specific operating system because it runs "outside" the game in a privileged way to try and detect cheating.
So they have anticheat on Windows, and their own consoles will have a different anticheat system that is specific for the console OS.
Running games on Linux via Proton is effectively an emulation or translation layer, and the Windows-specific anticheat is not going to work with that.
If Sony wanted to provide multiplayer support on Linux they'd also have to provide a native Linux implementation of the whole game, rather than relying on Proton, which sadly not many publishers are doing at all. So its technically quite understandable why this isn't possible.
Now, personally I think client anticheat is garbage and they should not be depending on that as a solution anyway, but that's a separate argument!