Yes, there are basically three ways to run Ubuntu:
- live-only booted from a DVD disk or USB drive
- persistent live booted from a DVD disk or USB drive
- Installed system (installed into internal drive or external drive (USB hard disk drive or pendrive or eSATA drive).
If I understand correctly, you are thinking about 'persistent live drive', and yes, you can create such a system in a USB hard disk drive (even in an internal hard disk drive), if you use mkusb. The partition for persistence will be created in the target drive (in the drive where the operating system is installed), but you can change that manually afterwards. The overlay system for persistence will search for a file/partition with the name/label 'casper-rw', and can find it in another drive too.
See the following links.
But you should also consider 'Installed system in a USB hard disk drive' according to the following link and links from it,
Boot Ubuntu from external drive
chroot environment
on disk, and then I installed kernel and grub, finally applied this patch https://github.com/lemonsqueeze/boot_chroot . Now I can boot into a directory which contains a linux distribution. I also managed anather way in Ubuntu 14.04 https://askubuntu.com/questions/405735/installing-multiple-linux-distros-in-a-single-partition/409229#409229 – kenn Jan 10 '18 at 19:39