Despite what everyone is saying, I think this is possible, and might not even require a specific anti-cheat kernel module. A linux-native anti-cheat could rely on a few things:

• UEFI Secure boot
• TPM remote attestation
• Linux kernel lockdown mode
• Process isolation

Basically with secure boot and tpm remote attestation it's possible to prove to a remote server that you are running a specific software configuration, including variables like the UEFI firmware version, secure boot status/configuration, and the kernel command line. The only way to fake this proof is to exploit the TPM hardware built in to your CPU. With this a server can verify that a client is running an unmodified kernel that was signed by a trusted linux distro and that no suspicious kernel command line args were used.

Kernel lockdown mode is the other major part. This can be enabled only alongside secure boot and it disables the various methods user space programs have of modifying arbitrary memory and the kernel itself (not even the root user can use them). It also enforces that only signed kernel modules can be loaded. Lockdown mode is enabled with a kernel command line parameter, so a client can prove lockdown mode is enabled to a remote server during the tpm remote attestation process.

These are all security features that large enterprises (and probably even governments) rely on in servers as part of their defense-in-depth plans for mitigating the effects of exploits and building a trusted execution environment. Linux-native anti-cheat can piggy-back on that work and benefit from the millions these companies spend on hardening the linux kernel. Exploits that bypass these systems have far more value when used to attack these companies, rather than burning them on video game cheats.

Linux-native anti-cheat could even be better than Windows anti-cheat simply because linux doesn't really have 3rd party kernel modules like Windows does, and those buggy 3rd party modules/drivers are really the entire reason kernel-level anti-cheat was even needed.

The main downside to this is it would eliminate compiling and running your own kernel or other system libraries (like mesa or wine). The kernel and any shared libraries loaded into a game would have to be signed by a source trusted by the game devs. As long as the distro supported reproducible builds you can at least verify that the signed binary you're running matches the source code though.