NT SERVICE\MSSQL$SQLEXPRESS user cannot be found when added to network drive
When you map that S: drive to a network share, you are using your local Windows account to do it (the user you're logged in as). However, SQL Server Express is running under a different account.

There are a few things you can try to work around this.


Create a network share which has "Everyone" permission to write to, and then use the UNC path (\\server\path) as the target inside SQL Server for backups.
Run the SQL Server service under a different account, which happens to coincide with an account using the same name and password on the network resource.
Back up your databases to a local drive, and use some synchronization method (ROBOCOPY, XCOPY, etc.) to move those files to the network share, running under your normal Windows credentials. You can achieve this using the Windows Task Scheduler, running a PowerShell script or a Windows batch file.