Although this is true for most games it is worth noting that it isn't universally true. Usermode anti-cheat does sometimes work verbatim in Wine, and some anti-cheat software has Proton support, though not all developers elect to enable it.
Looking at you Rust.
Edit:
And the rest of you. If even Microsoft's Masterchief Collection supports it, I Don't understand why everyone else does not.
It's because the Linux versions of those anti-cheats are significantly weaker than their Windows counterparts.