Is there a way to change the system date/time settings to a non-Gregorian calendar, e.g. Hijri, Jewish? Is there a way to change the internal Android system calendar to a completely different one?
No - Individual Apps have to implement and support the different calendars.

Internally the Android System kernel really only knows about UNIX Timestamp. If you change the date/time manually without any network, you'll be changing that value.

It is up to the Android Framework or third party developers to support the various calendars.

From the beginning of Android there has been the Gregorian Calendar class and only with Android 7.0 (Android API 24) and later have additional calendars like Islamic and Chinese been available within the Android Framework. You can see the other calendars available to developers in the Internationalization Base Calendar Class

Regardless, Android developers would have to add in their own code to support different calendars and related issues with the complexity of date/time math.

So sadly you'll have to check each app that you use to see if it has the internationalization capabilities you want.

If you wanted to change the screen lock/status bar/notification shade calendar information, that would probably be a Custom ROM which would expose the Calendar type to use in those fields.