The wildest one I've seen is Chrome on Android not allowing screenshots in Incognito mode. I'm struggling to understand the rationale there except maybe to prevent people from accidentally screenshotting sensitive stuff like porn or domestic violence resources? Either way, it's really not something the app should have control over.
I'm not sure why you're being downvoted. You're 100% right. Android lets apps block screenshots with FLAG_SECURE, and iOS does too with https://github.com/JayantBadlani/ScreenShield, and neither one gives the owner of the phone a way to override that.