How can I get my backup icon back in the Finder?

Editing that user pic should be straightforward there in System Preferences. Hover the cursor on the image and click Edit.

You might have to quit System Preferences after you edit the image so that it appears the next time you recall sys prefs.

If that doesn't work, you might try these old standbys...

If rebooting your Mac doesn’t fix things, see if the problem still happens in Safe mode (hold the shift key down at startup). It can take much longer to safe boot (10 min) so be patient.

How to use safe mode on your Mac - Apple Support 

Safe mode forces the OS to clear certain caches and logs and do other miscellaneous housekeeping. It's non-destructive to user data so is safe to do. Very often certain weird software behavior is corrected by simply logging in this way as a troubleshooting method. When you're in safe boot, the machine will not be at its best performance, especially with graphics, but that's expected.

Exit safe mode by restarting your Mac normally and re-evaluate the issue again.

Reset the NVRAM

How to Reset NVRAM / PRAM on your Mac - Apple Support

Reset the SMC

How to Reset the SMC of your Mac - Apple Support

Go to Finder > Preferences > General tab and check mark the box to Show External disks on the Desktop.

Apple has released several security updates recently, including one for Catalina. Maybe that's what happened, your Mac finally got caught up.

Seems strange. A head scratcher.

You're welcome. Good luck. 🍀