If you have any pauses, even fairly brief ones, in the output of the serial data that would give the receive time to re-establish synchronization, that it may not get if both devices are keeping their serial UARTs constantly busy stuffing words down the wire one after another with no gaps. Note that Arduino serial output is buffered, so typically a "noisy" program can "run ahead" generating data, but once the buffer is full the output calls will start blocking and the program will essentially run at the pace that keeps the serial output constantly busy.
Your multiplexor likely involves a fair amount of customization, which is yet another thing to spec / make, so there could be a lot of logic in giving each board its own unique channel.
You could just buy two inexpensive USB serial adapters. Distinguishing them by physical USB port is typically possible but can indeed be a bit painful. There are writeups around the web, perhaps some even here, and of course it depends on the host system.
If you get two different chipsets, it may be quite a bit easier to tell them apart. One modern CDC/ACM type Arduino and one legacy FTDI gets easier still, since on Linux or OSX these get device node names of quite different form.
You can also get a dual or quad channel converter, like the FTDI-2232 or FTDI-4232. These are typically more expensive than two single channel modules, but are handy gadgets in their own right (they also talk synchronous protocols like JTAG or presumably AVR ISP, alas, while possible trying to get one to be both a serial port and a JTAG adapter at the same time can put you back in the realm of driver battles - but I digress)