Attempted to install Catalina in Target Disk Mode.... bricked my mac?
What many people fail to take into account when they install a new version of macOS on a Mac, is that the process does not simply involve copying files to the Mac's internal hard drive, but also a firmware upgrade to make the firmware/EFI support booting from the new OS. This is why you get the message you get: if you boot a Mac in target mode and attempt to install a new version of macOS onto it, the Mac you install into would not boot (if you had managed), because it would never have had the firmware necessary to boot installed. 

For instance, when installing High Sierra on a Mac, the firmware necessary for the boot loader to recognise and boot from an APFS volume and disk structure must be installed, since HS was the first version of macOS to be able to boot from an AFPS volume. Similarly, when installing Catalina, the firmware necessary for the boot loader to parse the new two-volume read-only / read-write setup must be installed on the Mac that will boot the installed OS

So in short, the only viable way to be able to install (some versions of) macOS on a Mac, is to actually boot that Mac one way or another and install the version of macOS from the it, from the Mac that will run what you install.

In your case, if the only problem is a broken display, a viable solution is to boot the Mac with an external keyboard, mouse and display plugged in, while connected to a power adapter, since that should put the Mac in clamshell mode and fully operational. If you have already wiped the internal disk, put the working Mac into target mode and boot the broken Mac from the OS installed on the working Mac, plugged in via thunderbolt or Firwire (depending on connectivity options).