This is under System Settings > Privacy. Even if turned off, various apps still seem to remember what I opened last time I ran them. I thought that they might be caching the history in their config file, but if I purge recent files from this same menu, then the apps forget. I can't say that the setting doesn't work under some circumstances, but I've never seen a case where it does in fact work.
This is specific to Recent Files, typically accessed via Nautilus or a file picker. It doesn't necessary pertain to specific applications and what they may remember. By all accounts, the functionality is working as it's expected to. If you're expecting the functionality to be broader, then that's an upstream issue, given we just use Gnome Control Center.