This behavior seems eerily like phishing pop ups. I say this because I was unable to log into my email and suspected I'd been hacked. I recovered access and changed my password. A few hours later a request for my password came up again. I didn't respond to it and found my email to still be accessible. I suggest when these popups appear that you ignore them and see if your email updates. If you can't get it to update, use your email recovery procedure and don't respond to these popups in the future.