I keep getting an apple ID pop-up to verify the password and whatnot of a different person's email. How do I get rid of it so I can finally use iMessage?

Try and go to settings>messages to see if your iPhone and your Apple ID are stated there

If not have you bought this phone second hand? Maybe the old users' is still registered to messages

after that try this

• Try to restart your iPhone or iPad.
• Check your network connection. To send a message as iMessage or MMS, you need a cellular data or Wi-Fi connection. To send an SMS message, you need cellular network connection. If you turn on Wi-Fi Calling, you can send SMS messages over Wi-Fi.
• Check with your carrier to see if the type of message you’re trying to send, like MMS or SMS, is supported.
• If you’re trying to send group MMS messages on an iPhone, go to Settings > Messages and turn on MMS Messaging. If you don't see an option to turn on MMS Messaging or Group Messaging on your iPhone, then your carrier might not support this feature.
• Make sure that your device has enough space to receive images and videos.
• Make sure that you entered the correct phone number or email address for the contact.

After that

Contact Apple Support if you still can't send or receive messages

See this link for further reference

If you can't send or receive messages on your iPhone or iPad - Apple Support

I had something similar - Apple ID pop-up to verify password of redundant Apple ID email address continually appearing. I returned the iPhone 12 to factory settings and downloaded the apps one by one. I now have a clean phone with no verification pop-ups - bliss.