Some install permanent rootkits that are completely uninstallable (see: GameGuard‡, XIGNCODE⸸, AhnLab Hackshield†)
⸸ God awful basically malware. Sends a ton of data home to bare IPs overseas. No TLS. Incredibly invasive.
† This one was particularly interesting, malware could use the rootkit to hide their files based on specific names. It is horrendously shittily written. AhnLab itself is also interesting, as the founder tried to run for president of South Korea or something, and SK either still does or used to mandate some of their "anti keylogging rootkit" software as required by law. Sends a ton of data home to bare IPs overseas. No TLS. Incredibly invasive.
‡ This does not uninstall after you uninstall the game, it runs persistently, and would make windows non-genuine or toggle on unsafe mode for unsigned force-installed ring0 drivers. Sends a ton of data home to bare IPs overseas. No TLS. Best part: did not sign version updates or demand they be higher. Cheaters could run a webserver on localhost with empty 0-byte signature detection files, and it would "patch" itself down to uselessness at launch and detect nothing. Incredibly invasive.
They often run when the game is inactive.
In the past these anti-cheat modules have caused compatibility issues or had security vulnerabilities, which is fun. DRM packages tended to be similar.
If I check my Services pane right now, I have Easy Anti-Cheat and BattlEye listed which are both anti-cheat services installed by steam/etc video games that run with System level permissions. IIRC PunkBuster also uses a system service.
Because these are system services, they stay installed even after you remove the game. There are ways to manually remove them of course but the moment you launch a game that uses them, they'll get reinstalled.
I'm pretty sure the anti-cheat is just a library/module/etc. that's installed with the game. I can't imagine that it's a completely separate program that you install (unless it happens to be the game launcher).