In version 6.5 of Veeam Backup & Replication, the drive letters listed in the Backup Browser are determined by reading the registry hive stored on the system drive. If the system drive is not backed up in the same backup job, other drives will not display with their drive letters as they are within the Guest OS of the VM. This occurs specifically if you backup a single drive of a VM, and that drive is not the system drive.
The following can be found in the VeeamShell.log after the FLR has launched the Backup Browser:
Error Failed to set letters for mount points
Error Unable to locate the VM's registry (System.Exception)
Error Failed to detect drive letters for mount points.