The issue is caused by mediapools which are not linked to any library but still contain tape media entries. This may happen when the library or standalone drive was exchanged, and the tape mediapools and media were migrated manually into the new library / standalone drive without cleaning up old entries accordingly.
The logs show issues such as follows:
***Veeam*** SRV: [timestamp]:<00000001> [VEEAM|ERROR] ExecuteNonQuery failed for Database 'VeeamBackup'. (Microsoft.SqlServer.Management.Smo.FailedOperationException)
***Veeam*** SRV: [timestamp]:<00000001> [VEEAM|ERROR] Cannot insert the value NULL into column 'media_pool_id', table 'VeeamBackup.dbo.Tape.tape_mediums'; column does not allow nulls. UPDATE fails.