I have a fresh SuperDuper backup of Mojave and booted to it. Then I tried to install Catalina. The process completed and I booted into the external HD, but it didn’t appear to work. Is this not the way to install the OS on the external hardware?
I’m on a Macmini 2012 Server
Although I haven’t tried installing Catalina release over Mojave on an external drive, I had been running the beta that way – using Superduper to make an external clone, booting to the external, and then upgrading to Catalina (beta) on that.
When the process completes (at least in my case) it leaves me at the login prompt of Catalina. If you have the login prompt for Mojave maybe it booted the internal? Anyway, from Mojave you should see the mounts for Catalina and Catalina - Data on your desktop if it actually did the install.
It should not be a problem, did you check whether on reboot you are booting into the external disk?
After it completes the install part on the external disk, I held down the Option key. I chose to boot into the external disk that started the Catalina install and it just acts like it didn’t install anything. So I tried again, this time after it needed to reboot, I let it go instead of holidng down the option key, that seemed to be the ticket. At least for now…
Thanks for the confirmation of the process @tomalmy and @JKoopmans
Good luck 
Yep. Don’t hold down the Option key. In fact I’ve noted that at least on my new iMac you have to select the boot disk from System Preferences. The Option Key no longer works. Probably a security measure.
I think if the keyboard is wired with a USB cable, it will work. Been a minute since I tried it.
Mine is wired (USB). It worked on my older 2014 iMac (and older systems) but not on my new 2019 iMac.