As you only posted this a week or so ago, I thought I would tell you of a similar experience that my wife experienced.
Yesterday afternoon she got the same message as you suddenly pop up on her iPhone 13 for no reason at all. However she could still get her emails on our iMac.
I noticed that in addition to that message, at the bottom of the iPhone screen was the additional text “CANNOT GET MAIL The mailserver imap.******.com is not responding. Verify that you have entered the correct account info in Mail Settings Server code: AUTHENTICATION FAILED Server message “Authentication failed.””.
At first I was worried about malware because she had received an unsolicited email entitled “ALERT…” and going on about her account been nearly full up or something. I deleted it straight away, but I couldn’t be sure if her iPhone had not been compromised.
It took me about an hour to track down the problem. One clue was that we use POP3, not IMAP. I went into ‘Keychain’ on our iMac. Luckily I annotate passwords in Keychain so the first Apple password with my wife’s name next it did the trick. I entered it at the mysterious prompt and she can now access her emails on her iPhone again.
We have no idea why the password was forgotten by the iPhone in the first place prompting the message, .