Well I can't say exactly what is going on. However, one thing I've noticed with the new tools (9.1) is that in the Run Configuration dialog, you sometimes have to click on the "Refresh Connections" button (or something like that) so that it goes out and rescans the devices out on the JTAG chain and the corresponding JTAG debugger instances within those devices. Then you can select the device. Try that first and see if it resolves your issue.
You are correct that the system ID is compiled into your FPGA image. It does not / cannot change unless you regenerate your SoPC system and then recompile your FPGA image.
One thing you can do as a sanity check when the error occurs is look at what it's reporting as the system ID that it read from the device. Then look at your software files "system.h" inside your BSP folder and see if it matches. You can also check to see that it matches the ID listed in the Run configuration dialog.
Jake