Connecting Logitech Gaming Mouse - Not Connected in G Hub
After a few frustrating hours I found the solution.

You need to install the Logitech Control Center as well. Once you've installed it, restart your machine and it should now show up as expected in Logitech G Hub.

Notes:


If macOS discovers it as a keyboard, close the window and do not try to configure it as a keyboard.
If you configured it as a keyboard as above, you can try running this to reset it: sudo rm /Library/Preferences/com.apple.keyboardtype.plist
If you're using Karabiner-Elements, this might register it as a keyboard as well, causing issues. Open Karabiner -> Devices -> Basic Configuration and untick your mouse in that list.