I’m going thru the same thing. I get prompted on my iOS tho. It happens now and then and before this most recent time I always just entered in the key not thinking someone could spoof my settings app in my iPhone most recent model and fully updated iOS software. I didn’t like it before soliciting for the key when I made no apparent changes. The prompt comes out of left field. Im waiting for a call back from an Apple senior something of the other engineer or specialist. Didn’t commit that to memory. Anyways. They escalated it from the get go up to that person. Maybe it’s a big deal. From my curious nature I’ve deduced it maybe something to do w keychain synching. I have many devices i log in and out of that are running diff Mac OS’s on the same computer and I have three I use currently that are being logged into and out and the system being reinstalled….. don’t ask…. I fiddle a lot…. Can’t learn anything until it’s broken and have to fix it. Anyways. Possible reasons for unexpected recovery key verification prompt:
-you have a device that’s running an old operating system that doesn’t support modern security features. Or the operating system is too old to be trusted and secure w your synched data requiring you to delete device from your trusted list. I had that happen w my old iPhone 5s I think it was.
-again these are guesses…. You sync up the key but not a recovery contact (in case you lose the key)…. I had mine enabled but then I deleted that contact after I regained access to my account way back. May have made the system go wonky or something by deleting the contact… So I just added a recovery contact. Still the prompt remains. Which I expected. This was more of a maybe it will stop prompting them from here on out after I re verify my key. I Don’t know. Just a thought
-maybe all the devices you have trusted need to be set up individually to be able to use the recovery key system. Reading something about how once you set up a recovery key the usual methods of account recovery no longer work thru Apple support. Or some ish. So if you set up the key w your Mac and you also have an iPhone maybe you have to go thru the process on the phone since it’s in your trusted list of devices.
-it could also be a design feature to make sure you keep your key somewhere attainable and not in the trash forgotten bout…. Bit of a stretch but I don’t know, there’s wonkier things that go on w synching data across devices and keeping it secure.
Other then the above well thought out and researched longer then 1 hour solutions then def most likely Sasquatch did a brute force attack on your password and added his new Apple Watch to your account just to show that his intelligence is superior.
but no just kidding. Once I get a hold of the Apple dude who will most likely tell me to just insert the recovery key and see if it happens again I’ll let you know what if anything is said to help if you’re still encountering the problem. 
i have a recovery key set up and I know the key. But I get these same above pics about monthly intervals id say. And during times when I havnt updated any settings on my Apple ID or cloud or keychain. Or logged into a new device. Or logged out or reinstalled any Mac OS or iOS or restore from a backup or even fart out an Apple and even tho I didn’t provoke a prompt every now and then i see that darn red number one in my settings app. And guess what it’s telling me. Verify your recovery key or account castration. W a dull rusty used tool. Dipped in Covidness. So naturally I always typed it in. But this time i decided to get frustrated waiting on apple to call me back before entering my key.
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