A number of Gigabyte boards from ~2009 have broken BIOS support, resulting in the PHY reporting an invalid PHY ID. Realtek / Gigabyte don't release errata information, therefore there's not much that can be done. In bugzilla.kernel.org you can find workarounds that helped for some users, else use the r8168 driver.
In https://bugzilla.kernel.org/show_bug.cgi?id=207203 it was suggested to enable the boot ROM in the BIOS and/or reinsert the r8169 kernel module. Neither of these worked for me. Fortunately the r8168-dkms package does work. Thank you for the suggestion, as I was not aware of this driver.
Roger