To prove that this is a metric, consider the following theorem.
Theorem. If the second player can survive for $n$ steps in the $(\Gamma_1,\Gamma_2)$ game, and for $m$ steps in the $(\Gamma_2,\Gamma_3)$ game, then he can survive for $\min(n,m)$ steps in the $(\Gamma_1,\Gamma_3)$ game.
Proof. The idea is simply to combine the strategies for the two games. Fix strategies for the second player in the $(\Gamma_1,\Gamma_2)$ and $(\Gamma_2,\Gamma_3)$ games. In the main $(\Gamma_1,\Gamma_3)$ game, now, let the second player answer any move of the opponent on either side by first copying the move into $\Gamma_2$, and then copying the response of that move into $\Gamma_1$ or $\Gamma_3$, accordingly. In this way, every play of the $(\Gamma_1,\Gamma_3)$ game can be seen as the composition of the strategies for $(\Gamma_1,\Gamma_2)$ and $(\Gamma_2,\Gamma_3)$ games. Since the resulting finite subgraphs $A_1$ and $A_2$ will be isomorphic for $n$ steps and $A_2$,$A_3$ isomorphic for $m$ steps, we will thereby maintain $A_1$ isomorphic to $A_3$ for $\min(n,m)$ steps, as desired. $\Box$
It now follows that the distance from $\Gamma_1$ to $\Gamma_3$ is no larger than the shortest distance from $\Gamma_1$ to $\Gamma_2$ or $\Gamma_2$ to $\Gamma_3$, and so we will obey the triangle inequality.
Your games are closely related to Ehrenfeucht–Fraïssé games, but not exactly the same, since in the Ehrenfeucht–Fraïssé games, one keeps track of the moves, and these must be the isomorphism, but the way you described the game, it seems that you allow one to change the isomorphism as play proceeds.
The argument I gave above is the analogue of Lemma 3.2.1(c) in Hodges Model Theory. He shows that Ehrenfeucht–Fraïssé games determine an equivalence relation for length-$\gamma$ play, and this is obtained precisely by copying moves into a supplemental game as I described.