OK, I think I might have identified where the problem is and a workaround (that seems to work for me at least)…
In my case, the bug happens when I have a dock with an attached monitor plugged in when I attempt to start Plex.
To workaround the problem, unplug the dock (you must unplug the dock) and directly connect a second monitor to the Mac (via HDMI or TB). Start Plex, it should come up on the second monitor. You can then drag Plex onto your first monitor. Reposition the window on the first monitor a couple of times (to make sure that the saved window position is definitely on the first monitor), then quit. Restart Plex and should come up on the first monitor - so you can now unplug the second monitor if you want.
NOTE: This is a workaround and not a fix because as soon as you go back to using your dock (with no directly connected monitors), the problem will come back. You can plug in the dock while Plex is running and everything keep working, but as soon as you quit and attempt to restart, the problem will come back. You can have one monitor via a dock and one monitor directly connected, that works, but if you’re using a dock in the first place then that may be of limited use to you…