The PS Vita could not be linked to the PS4 System for Remote Play as the accounts on the systems are not the same.

To use the Remote Play feature, the same account must be linked to both systems. 

If you need to switch the account on your PS4, select [Power] > [Switch User] on the home screen.

Only one account can be linked to a PS Vita. To link a new account, you will need to restore your PS Vita system.

You can restore the system software to its default condition and delete all of the data in system storage by selecting [Format] > [Restore This System], and then follow the on-screen instructions to complete the operation.