Use discord in the browser
Unfortulately, Discord does not provide any built-in option to do that. However, if you use the browser version, Discord will be confined to your browser's sandbox and won't be able to scan your computer for running apps, and most¹ other features should work properly.
@aphid's answer is generally correct, although as is now, the most convenient solution is hidden among many suggestions which are possible for power users, but overkill for most Discord users (deploying a VM, installing a VPN, reverse-enginerring Discord, etc). I'd like to put more emphasis on what most people should consider:
If you goal is only to stop Discord from finding out which games you play, as @PausePause pointed out, Discord itself does not provide any option for that. The closest that can be achieved is by running Discord in a browser, which will confine Discord inside a secure sandbox and will prevent it from exploring the rest of your system.
Note that this will also disable a few other features that also require access outside the sandbox¹. However, I've been using Discord in a browser everyday for over a year, and it works well, so it may be a viable solution for you. Best is to try!
¹: Certain features like push-to-talk/push-to-mute won't work if you don't have the browser window focused, but all important features (voice chats, screen sharing, etc.) should work fine.