I think you're wrong, but am willing to be told otherwise. Here's why: one currency is not (generally) legal tender outside its country of origin, so there's an exchange rate determined roughly by the trade balance between countries etc. If all currencies were legal tender everywhere (a much more analogous) situation, the exchange rate would not be determined by the same factors, since the currency would in no way be pegged to the nation.

Then the question becomes: what does in fact determine the exchange rates? I.e., would one currency be "infinitely valuable" compared to all others? Which one? The first? Or would all of the stabilize at the point where 1 of A is always worth 1 of B, because that's easiest to work with?