If you restored the new iPhone from a backup, or are trying to transfer data from an old iPhone through the direct migration it will ask for any Apple Id that exists in the old iPhone in any way.
Normally that happens when you have apps that were downloaded with the old Apple ID on the old iPhone and they are trying to be restored.
Note, that unless manualyl deleted though Apple's vfery specific Apple ID deletion process, Apple IDs never stop functioning even if the associated email address is no longer functional.
If you don't know the password for it anymore, you will need to attempt to recover it, though without a functional email, it may be impossible to get the recovery email.
Start here: If you forgot your Apple ID password - Apple Support
then contact iTunes Account Security if it fails: iTunes Store - Contact Us - Official Apple Support