I changed my AppleID email address months ago but after upgrading to Mojave...

If you click Cancel, does the request go away? Two possibilities.

I read a post a couple of days ago where someone posted they had talked to Apple Support about this and were told it is a known bug that Apple is trying to correct.

Check the App Store and iTunes Store to see if there are any updates for apps/purchases. As posted above, request for an old Apple ID password is often caused by updates to apps/purchases that were made when using the old ID. Those purchases are tied to the old ID and can't be transferred. One option is to delete the apps/purchases made by the old ID and download them using the new ID if desired.

This is normal. It is because one or more apps running on the iPhone were purchased using the old Apple ID and you will continue to have to sign in with each update to the software. The important thing is that your new Apple ID is showing correctly in Settings. When you click on settings the first thing at the top will be your profile, click on it. At the top you will see your picture with your current Apple ID underneath it. If you see the correct one, no further action is required. If you see the incorrect one, scroll to the very bottom and select "sign out". Once you have signed out, you can sign back in with the correct Apple ID