I launched the second copy of bitmessage from the same user on the same display, and it launched. It showed two windows and two tray icons.
This shouldn't happen. It should locate the first instance, make its window active and quit. Chrome browser, for example, does this.
The second instance also shouldn't be allowed to launch off the same database on another screen, since this doesn't make sense.
It is possible to find the window of the original instance through the window manager.