It sounds to me like you are looking to create a "portable" linux install.
This used to be quite common when linux was much less common.
A lot of live distros, including ubuntu, support using an extra partition on the USB as the 'drive', so you boot to the standard live distro but your files and history are all there too.
Here's a couple links. Google "portable linux" for more info.
A general list of portable linux distros.
https://www.ubuntupit.com/best-portable-linux-os-distros-to-install-run-from-usb/
Detailed instructions on doing this.
(I reviewed this, looks accurate, didn't really dig through it. Their method is to create an installer USB in one port, boot from it, and then install to another USB in another port. The UEFI directions at the end look a bit sus. YMMV.)
https://coastipc.com/blog/post/creating-a-portable-linux-operating-system-on-a-usb-drive