No. I've heard of the Scotch, King's Indian Attack, and some French opening. And also the Sicilian.
This guy keeps copying my same exact moves!

u should just take a piece. get him in check.
He's most likely going to castle next move, so I'm going to take that pawn to open up his kingside.
Excuse me, it is minor, but this discussion is illegal during an on-going game.

First of all, what strikes illegal is the fact that some posters actually comment on the game. The OP has not required you to do this.
About the original question: There is no rule in chess which forbids to play symmetrical defences. At some point it may not be practical for the defence, but it's not forbidden.
However, what I understand to be prohibited is copying your opponent's moves when you play two games at once with reversed colours, for example in team matches where both games start at once.
I hope this helps a bit...

make a move that can not be copied, like a pawn capture...or even move your king one space...that'll teach him.

If I ever play you im going to copy your moves just because I know its going to drive you nuts and maybe even throw you off your game. Miami is looking good again.

I haven't had someone do that to me for ages. i would love them to copy me because i can use that knowledge against them so they eventually cant copy it, because there in check or some other nasty attack they are forced to defend against and breaks there position fast. Just find away that you can disrupt that plan and perhaps increase your own while ur at it.
Ask yourself how can i take advantage of the fact they are copying me? how can i play move that will help me if they continue to copy my moves?

Nice one tonydal! The furthest someone has copied my moves in the symmetrical English has been 4 moves. Such a disappointment. Maybe my moves just aren't good enough to copy too far.
He didn't copy your moves, you went Nf3, he went Nc6. It's a doubled king pawn closed position, they are often very very symmetrical. I don't see what the big deal is? Out of 6 moves, 4 were the same, not even necessarily at the same time. That is normal, ever heard of the 4 knights opening??