Why won't my Mac mini stay in 24 hour time?

Give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

A SafeBoot Start up your Mac in safe mode - Apple Support will sort many anomalies

Login and test. Reboot as normal and test. Caches get rebuilt automatically.

To be proactive you can file a bug report / submit your Apple Feedback here: Product Feedback - Apple

Try turning off AM/PM manually in the Control Center settings(?)

no resolve— try turning off "automatic"...

I would reboot and compare your results.

If you still have an issue

from the Terminal.app copy & paste:

sudo rm /var/db/timed/com.apple.timed.plist

this will remove the plist. and rebuild anew.

(note—your psswd will not echo on screen type it in anyway, use the enter\return key to proceed. Quit Terminal.)

I seem to remember a similar case some time ago where the solution, unexpected as it may have been, was

to ALSO turn on 24 hour clock on "Lock Screen" - your line just below the marked red screen: