I keep receiving the message “verification failed. The application you selected does not exist” whenever I am prompted to verify my ID. I even tried changing my password.

There was an Apple server outage last night. Have you tried again?

Do you have access to another WiFi network where you can try? That has worked for several people.

Try a restart.

Restart your iPhone, iPad, or iPod touch         

Try a forced restart.

Reset Device.      

If necessary

Hard reset.        

Do a backup and try a factory reset. Then restore from your backup.

Restore to Factory setting.      

Backup - Restore your device from an iCloud or iTunes backup.      

There are so many similar problems related to connecting to an Apple server that I have ceased trying to sort them out. I suggest you do this search (verification failed apple server) or use my search link verification failed apple server. You will find quite a few reads that may be related to your problem. You also may want to use the support options below in dealing with your specific problem.

Get a call from Apple support to help solve your problem using this

link https://getsupport.apple.com/?caller=kbase&PGF=PGF63005&category_id=SC0245&symptom_id=23362

Apple ID Support https://support.apple.com/apple-id

Apple ID account security for country-specific and region-specific help https://support.apple.com/en-us/HT204169

Get Support: We’re here to help https://getsupport.apple.com/. Go through the options to find the best support option.

You are welcome.