I don't have a clear answer to your problem, but I'll highlight several things I've noticed:
all the directories have 777 permision (directories: asus, multimedia and bitcoin) to "media" directory I can't change permissions.
This is bad security, you should use something like 750 or 700.
is not an externa drive is just one 500gb HDD that I partitioned 100gb-for ubuntu OS- and 400gb-for bitcoin blockchain- (ntfs , ext4 I try with both but same error) see the picture
https://i.ibb.co/NpM2kyg/not-working.jpgDon't use NTFS, use a filesystem designed for Linux. Ext4 is fine.
the only way to work is if when I install Ubuntu I give the 400gb partition mount point "/home" but this is something else is not that I want cause will not show the partition like a device is just a folder.
Everything is just a file or folder on Linux. You could mount it as /home/asus/snap/bitcoin-core/common/.bitcoin if you want.
you can try it yourself without installing ubuntu you will see when you choose to safe the blockchain to a different location (a partition) will not work
Wait: did you install Ubuntu on your PC, or are you running Ubuntu LIVE without installation?