OK, here is the report from the 'guru' (apparently, that would now be me) -
A. - The problem is most likely that your backups are corrupt. LAMPP Backups ONLY work when you are logged in as root. Note the errors you got - they were generated by the next to last step here - the script '/opt/lampp/share/lampp/backup.head' which is tacked on (catted) to the backup archive in the next to last step (before the temporary files are deleted) Read through this next part, and you will be ready to debug:
B. - Here's how to debug your problem. You will need to run some commands from term/CLI. You will also need to understand (to some extent) the bash script(s) that are being run, the 'root/start' script would be /opt/lampp/lampp. Use an editor to peer into the script files mentioned.
/opt/lampp/lampp
Right around line 493, if the case is the parameter is 'backup', another script is called: /opt/lampp/share/lampp/backup. This is the workhorse, and location of whatever did not work during the backup, failed.
/opt/lampp/share/lampp/backup
The first part sets up the work and does a MySQL validation -
Lines 35-51 iterate the databases found, and writes each database out to a .sql text file.
Line 53 gets the current XAMPP version information
Line 55 stuffs these files and folders (/opt/lampp/etc /opt/lampp/htdocs /opt/lampp/logs /opt/lampp/phpmyadmin/config.inc.php) into data.tar.gz
Line 58 gets the checksum of (data.tar.gz VERSION *.sql) Static files + version-info + all the .sql files)
Line 61 stuffs the prior outputs into a new 'master' backup 'backup.tar.gz'
Line 62 concatenates/joins (cat) a bash header (/opt/lampp/share/lampp/backup.head)onto the 'master' backup file just created, so it can be run as a script later. All of that (the output of cat) is stuffed into your dated backup file.
Line 63 erases all the temporary files -- for debugging I would comment it out, and examine then the archive outputs to see if they are valid archives, and check the MD5 Checksum.
You should read through these files to look for your problems. try running the commands line by line and see where they error out.
OK, you have the roadmap now.
Since this worked perfectly on my my machine, that indicates you probably had problems with ownership, permissions, or user access (who you were logged in as taking the backups)
You all owe me a beer!