My Mac mini is asking to set up my iCloud account even though it was already set up

I see this regularly, not frequently, but particularly after a software update. In recent years I have learned to dismiss this alert and then check in System Settings -> Apple Account that iCloud is still enabled, and that the settings are correct. Everything ithere s always in order. If you do log in again etc., you have the alert on other devices that a "New Mac" (sometimes my 2013 MacBook Pro) has connected to iCloud etc.

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.

From where did you buy this "refurbished" Mac mini?

Only Apple sells a certified, Apple refurbished Mac and provides the warranty to go with it.

Any other place may not have properly refurbished and prepared your Mac for transfer to you, and in the worst case scenario this computer is compromised.

Does the Apple Account that is currently signed in belong to you and your own email account?

FWIW I occasionally get such prompt to re-log-in to my iCloud account after I have dual-booted to an external disk running same or especially older macOS version. My main macOS is Sequoia but I might boot to Mojave or some macOS in between. I guess iCloud then just wants to set things correct.