Why does my new bootable external SSD show up as 2 volumes? I have an iMac (model 18,2) running Mojave. I have installed a new external (USB 3.1) SSD, and installed Catalina on it (and am in the process of migrating to it as the primary load source).

When I am running under Catalina, both the new SSD and the internal hard drive show up as single volumes ("SanDisk SSD" and "Macintosh HD"). But when I am running under Mojave, the SSD shows up as two volumes: "SanDisk SSD" and "SanDisk SSD - Data." Moreover, when I put custom icons on those two volumes (via Get Info) from Mojave, then reboot to Catalina, it's the icon I applied to "SanDisk SSD - Data" that appears.

This is most confusing. Can somebody shed any light on what's going on?

(FWIW, my customary disk drive icons are based on a photograph of a 1960s-vintage removable disk pack, of the sort that might go into a "Merlin" drive on an IBM 370 or compatible.)
You are seeing Catalina's way of setting up boot volumes, that's all. In Catalina, when you install macOS on a drive, it gets split into two volumes behind the scene: one read-only where the system and all system files reside, and one writable where your user files and every file that must be writable resides.

This is a way to further harden and protect the system from any accidental or malevolent change. The split is done by some file system level 'magic' and the volumes occupy overlapping space on the physical disk.

The preceding is a bit simplified, but you get the idea. When you see this volume in Catalina, it is presented as a single volume to the user. But when you see it in Mojave, it is shown as the two volumes it appears to be for Mojave, since Mojave has no knowledge of Catalina's boot disk setup scheme.

To use under Mojave, you can just ignore the volume without the "- data" and use the volume with "- data". All changes you do to that volume (as you have already seen with the icon) should show up back in Catalina.