Read this short description before answering please :). TID and SID values are random, but still have a 1 out of 4,294,967,296 chance of being the same across two different games. If the game you caught the shiny faraway island mew on has a TID and SID matching the TID and SID of an English game, could you re-name it to "Mew"?
Also, if anyone cares enough to find out, is it also possible for it to be traded up through Bank? I'm more curious about the top part more than anything else, so don't feel obligated to answer this part.