Installed Ubuntu is portable
Yes it is possible to install Ubuntu in one computer and move the drive to another computer (or to use an external drive) and run Ubuntu in another computer. It will often work (but not always, the hardware might be too different).
You should avoid proprietary drivers (for example for graphics and wifi). The following link and links from it might help you see what to do and what to avoid,
Boot Ubuntu from external drive
In principle, you can install Ubuntu 'as usual', except that it can cause problems, if other drives (hard disk drives or SSDs) are connected during the installation, particularly in UEFI mode.
The installed system will be created for the boot mode, in which it was created. So if you intend to run your target computer (the laptop with a damaged screen) in UEFI mode, you should boot the other computer in UEFI mode and install when running in that mode. And boot in BIOS mode (alias CSM alias legacy mode) to make the installed system if you intend to run your target computer in BIOS mode.
If a proprietary driver is necessary, it is best to install it after the drive has been moved to the computer, where it is to be used. In order to prepare for proprietary graphics, you may add the boot option
nomodeset to the default menuentry before moving the drive to the target computer.
Persistent live system
Alternative, if the computers are too different, so that a system installed in one computer does not work in the other computer:
A persistent live system
- is more portable (boots and works in more computers)
- causes less wear of the drive
- is not as stable (is easier to damage)
- can not be completely updated & upgraded (compared to an installed system)
See this link,