Having been very happy with 14.04 I recently installed 16.04 in a new partition (/dev/sd9). Both share /home (/dev/sda6) in a separate partition so I will not be losing my data if I delete 14.04 partition (/dev/sda5) . However will Grub2 be OK with this and will I be able to continue to boot 16.04 successfully.
The grub boot loader lives as I can see from the photo you've provided on sda1 (EFI system partition). Unless you manually uninstalled grub from your 16.04 install or expressly chose during installation to not install grub at all you should be fine, provided you copy your 14.04 /boot/grub/grub.cfg to your 16.04 install (same directory) or generate another with
update-grub (from 16.04).