Where does Safari Web Inspector store images?
This indeed looks to be a bug.

You can watch all file access on your system with the opensnoop command. This requires root access - run it, then try hitting the download button - you'll see no access from Safari.

Time to hit Feedback Assistant and tell them what's going on!

The earlier comment is a good workaround - once the image is displayed, just hit CMD-C to copy it for pasting elsewhere.