Installed Ubuntu systems are rather portable between (intel/amd) PC computers. Not as portable as live systems, because the live systems have a mechanism to try more drivers to critical hardware, for example graphics and wifi.
If you avoid installing proprietary drivers, and the computers work without proprietary drives, you will be able to install Ubuntu in one computer and run it in another computer. My son has an Elitebook 8560p, which is similar to yours, and it works with Ubuntu without any proprietary drivers.
In this case you should also consider the issue of UEFI versus BIOS. The Elitebook is probably booting in BIOS mode, while the computer with the Asus motherboard might be booting in UEFI mode. And you should install Ubuntu in the mode that you intend to boot it - or use an installed system, that boots in both UEFI and BIOS mode. I made compressed image files of such systems, that you can install with mkusb according to the following link,
Use one of these compressed image files, or use the link only to help you create your own installed system from an Ubuntu iso file.
If you create your own system, it helps to remove the internal drive (with Windows), so that the bootloader will written to the correct target drive (the drive with the USB - SATA connection). This is particularly important if your computer is booting in UEFI mode, but as mentioned before, if the Elitebook is booting in BIOS mode, you should boot your computer with the Asus motherboard in BIOS mode, when you install Ubuntu.