Google Play Services only recognizes deleted account; cannot update any apps
This could be happen because you deleted your account permanently from the web but that account still saved in the device and triggering a force sign in to identify you.

Follow these steps: 

For Samsung specific-


Open Settings》Cloud and accounts 》Accounts》Google.
Now click the account that you have deleted permanently.


Tap 3 dots on the upper right corner and choose Remove account.



It must work.

Else last option would be a factory reset and set up device again.