"Ignore ownership on this volume" deselected each time external HDD is mounted

I found answer to my question - yes, it is possible to modify ownership for the disk or partition with Terminal app diskutil.

Man page paragraph regarding enabling ownership explains:

Enable ownership of a volume. The on-root-disk Volume Database at /var/db/volinfo.database is manipulated such that the User and Group ID settings of files, directories, and links (file system objects, or "FSOs") on the target volume are taken into account.

This setting for a particular volume is persistent across ejects and injects of that volume as seen by the current OS, even across reboots of that OS, because of the entries in this OS's Volume Database. Note thus that the setting is not kept on the target disk, nor is it in-memory.

For some locations of devices (e.g. internal hard disks), consideration of ownership settings on FSOs is the default. For others (e.g. plug-in USB disks), it is not.

When ownership is disabled, Owner and Group ID settings on FSOs appear to the user and programs as the current user and group instead of their actual on-disk settings, in order to make it easy to use a plug-in disk of which the user has physical possession.

When ownership is enabled, the Owner and Group ID settings that exist on the disk are taken into account for determining access, and exact settings are written to the disk as FSOs are created. A common reason for having to enable ownership is when a disk is to contain FSOs whose User and Group ID settings, and thus permissions behavior overall, is critically important, such as when the plug-in disk contains system files to be changed or added to.

And command 'sudo diskutil disableOwnership device' disables owners for the 'device' which can be any version of the device identification:

- The disk identifier (see below). Any entry of the form of disk*, e.g. disk1s9.

- The device node entry containing the disk identifier. Any entry of the form of /dev/[r]disk*, e.g. /dev/disk2.

- The volume mount point. Any entry of the form of /Volumes/*, e.g. /Volumes/Untitled. In most cases, a "custom" mount point, e.g. /your/custom/mountpoint/here is also accepted.