@LukeAlmighty my log-in screen just goes to whatever one is recognized as the primary monitor and turned on. if it's turned off it'll just jump down the list of whatever is turned on.
in linux mint i can see it displaying on all windows and the log-in prompt follows the mouse. i don't know if you can actually change what display is the main one right from the log-in screen though.
i'm guessing if you are wanting to do something like that on windows, you might be able to by using a workaround to launch the cmd/powershell on the log-in screen. i'm sure there is a command for doing that.
@LukeAlmighty so you gave two PCs plugged into one monitor. There should be a setting on your monitor to disable automatic input switching. I don't think that's a windows problem, if I understand correctly. It's just that windows will send a signal to a monitor that is sleeping or turned to a different input that it's time to wake up or switch. Consoles do this too. I think on ps4/ps5 you can turn that functionality off, but you can also do it on most monitors/TVs.