How can I change backgrounds on different Mac desktops?

Upon closer inspection, this is old code intended for Intel platforms, with many framework changes since it was last developed, and would be a pain to build in Xcode. The homebrew installation would require Apple's Rosetta2 byte transcoder and no guarantee even that would work.

Update to macOS Sonoma 14.7.6. The ability to do that is on your System Settings > Software Update panel. If your mini maxes out at Sonoma, just apply the update for Sonoma 14.7.6. If there is an upgrade to Sequoia present, the scroll down until you see "Another update is available. Click the blue More Info… link and it will allow you to select Sonoma 14.7.6 and only install it.

It is possible using a couple of Apple Classes (NSScreen, NSWorkspace) to get a list of screens and set specific images for each screen. Unfortunately, I only use a single 32-in display on my M4 mini, don't have a spare screen to configure, and won't be able to test any currently unwritten code to achieve the first sentence.

I am running Sequoia, and using two displays. AFAICT, there was no change from Sonoma, so it should work for you also.

I can have different images on different displays, and different desktops in each display.

It is possible that these may at some point be lost; for example, if you accidentally closed a desktop and then added a new one, the new one won't magically inherit the previous image.

In System Settings->Desktop & Dock, under Mission Control, you may want to turn off "Automatically rearrange Spaces based on most recent use"

Also, when shutting down your mac, make sure that "reopen windows when logging back in" is checked.

Would you be satisfied with the numeric Desktop number clearly visible on the menu bar, rather than custom desktop images with a number to keep organized? See whichspace example.