There are basically three different configurations of Ubuntu systems in USB drives.

• Live-only system for testing, installing, repairing


• Persistent live system for a portable system, where you can install programs and save data


• Installed system, installed like into an internal drive, but into a USB drive

Live-only system

• Easiest to create, simply clone from the iso file to the USB drive


• Good enough in most cases


• Boots both in UEFI mode and BIOS mode

Persistent live system

• Fairly easy to create, when you use a suitable tool


• Very good portability between computers


• You can install application programs, tweak the system and save files


• You cannot upgrade the linux kernel and its device drivers


• Sensitive to corruption (you need frequent backups)


• Boots both in UEFI mode and BIOS mode

Installed system

An installed system, installed like into an internal drive, but into a USB drive.

• Fairly easy to create, if you want to run in either UEFI or BIOS mode


• Possible but more difficult to make it boot both in UEFI and BIOS mode


• Fairly good portability between computers


• You can install and tweak 'everything' just like with a system installed into an internal drive (only limited by the size of the drive)


• Fairly robust to corruption, but a USB pendrive's hardware is sensitive to corruption (you need frequent backups)

Details and links

I think that you want an installed system, that can boot both in UEFI and BIOS mode, and in that case the following links may be helpful,

https://help.ubuntu.com/community/Installation/UEFI-and-BIOS

How do I install Ubuntu to a USB key? (without using Startup Disk Creator)

Installed system booting via mkusb's boot system by C.S.Cameron