I thought you were wrong, and I wrote a reply, but then thought I’d test my guess (that Apple once again messed up the terminology).
I have 4 desktops on monitor 1. I just tested DEVONthink in the Dock (it is open on Desktop 2 monitor 1) and the choices are Assign to > All Desktops, Desktop on Display 1, Desktop on Display 2 (checked), and None (checked). I always have to move it to the correct desktop manually after a reboot.
Accordance is on my Desktop 3 monitor 1, but its choices are the same as DEVONthink’s; there is no “Desktop on Display 3” option. Is this a problem with the often-confusing Apple terminology? When they say “Desktop on Display 2,” do they mean “the current desktop where it is now open” on your right-hand display?
That seems to be the case! I just tested both applications. In the Dock, I checked “Desktop on Display 1” for DEVONthink, then quit it (on Desktop 2 display 1). I returned to Desktop 1 and opened DEVONthink. The screen swept over to Desktop 2 and DEVONthink opened there. I then checked “Desktop on Display 1” for Accordance, quit it (on Desktop 3 display 1). I returned to Desktop 1 and opened Accordance. The screen swept over and Accordance opened there. I swiped down to bring up the Desktop switcher, and Accordance (the current desktop) was #2 (uh oh), but DEVONthink’s desktop was now #3; Accordance was not on top of DEVONthink. So, except for arranging the order of my desktops, this might have resolved my issue. Thank you SO MUCH for this tip!
Apple Mail always opens on the right-hand display, but then I don’t traditionally have extra desktops on that monitor. It’s checked “None” in the desktop assignment submenu.