, an error message says I must have MOJAVE to install an update to MOJAVE. What am I doing wrong?

One can not install an Older version - Mojave - of macOS Over and Newer version - Catalina - of macOS.

Below Method will WIPE ALL DATA and there is no Recovery - Period

To perform this action will require booting from a Bootable Installer. 

This will have to be performed from a Qualifying Computer to run the version of macOS to be made on the Bootable Installer. Example : Bootable Installer of Big Sur would have to be done on a computer that Qualifies to run Big Sur.

Once that is done read on for preparing the Destination computer 

>> Only works on Intel Based Apple Computers.

Extra Special Notation regarding the Touch ID equipped Apple Computer

About Startup Security Utility and Must Enable from Recovery Mode the ability to boot from External Drive Before Attempting 

1 - Shutdown computer and disconnect all external drive Except the newly created Bootable Installer.

2- Restart and immediately hold the OPTION key until the Startup Manager appears and choose the USB Drive. 

3 - It will present options >> Disk Utilities >> View >> View ALL attached Drives. 

4 - Choose the Upper Most Drive ( not the volumes indented and list below ).

5 - The drive normally is called Apple Media or Apple SSD - that is the drive to Erase and format as APFS with the GUID Partition Map.

6 - Once that is done >> backup out of Disk Utilities and choose install macOS. 

7 - Follow the prompts and it may automatically reboot several time. 

8 - Upon a final reboot - Setup Assist will present with the newer version of macOS.

The installers from Apple's downloads area are updates, not full installers, and you need to get the full Mojave from the Mac App Store link in How to get old versions of macOS. As mentioned, use strictly Safari for this activity.

Ideally, you have a last Time Machine backup of the previous Mojave installation prior to the Catalina upgrade, and you did not reuse that TM drive for Catalina backups. If that were true, you could boot Catalina into Recovery and choose Restore from Time Machine Backup. That would wipe Catalina, and repave it with not only Mojave but your drive contents prior to the Catalina upgrade. It would be as though you never left.

Even if the second paragraph is true, I recommend that you make a separate Time Machine backup of Catalina on a different drive before you begin the downgrade to Mojave. Think CYA.

If the second paragraph is false, and all you have is a Catalina TM backup, then the bootable USB stick that you make from the full Mojave installer will allow you to wipe Catalina, and install Mojave. Once you reboot, and part of the Mojave setup process, you will be asked for a TM backup for migration purposes, and I honestly don't know how that will end. Catalina and Mojave use different Mail versions and when supplying a Catalina TM backup, you are depositing Mail content that Mojave may not know how to convert. Roll with it and see if my concern is wrong. There may be other Catalina-specific files that may give Mojave gas too.

Backup, download full Mojave installer from Apple with Safari, make a bootable USB flash drive from it, boot it, erase the target drive (which you backupped) as APFS, install Mojave, restore from backup.