Why does the copycat strategy work for two parallel chess games?
The setting in the link you gave (where is it referred to as $chess\lor \neg chess$) is that the player is playing on two boards with different colors, and has to win in at least one.

In that case you can win by mimicking your opponent as follows. Let $b_{\text{white}},b_{\text{black}}$ be the boards on which you play white/black correspondingly. Whenever the opponent makes a move $x$ on $b_{\text{black}}$, you play $x$ on $b_{\text{white}}$, similarly when your opponent makes a move $x$ on $b_{white}$ your play $x$ on $b_{black}$. The result is that you are playing the same position on both boards, only with different colors.