SQL Server Management Studio backup not shown when restoring
Normally when you want to restore a database from a different server,  you're looking to create a device backup to a network drive, and restore from that.

The way the restore from database interface works is by querying the msdb database for recent backups, to see what you can restore. Accordingly you should be using a device backup:


  If the backup is taken from a different server, the destination server will not have the backup history information for the specified database. In this case, select Device to manually specify the file or device to restore.


As the msdb database on server 2 will have no knowledge of the backups taken on server 1.