Updating FF caused the "Firefox is already running, but is not responding" error. But only on one machine.
Okay, so I have two machines - each of them running Firefox Portable Dev Edition. One was v119, the other might have been v118.
I updated both of them to v127. Afterwards, Computer-A runs fine, but Computer-B gives the "Firefox is already running, but is not responding. The old Firefox process must be closed to open a new window." error if you try to launch it again to open a new window (as opposed to Ctrl+N).