What you describe above is called a Cantor scheme in a polish space X. If one also requires that the closure of $N_v$ is a subset of $N_u$ whenever $u < v$, we get a point in X along each branch of this tree and the resulting map gives an embedding the the Cantor space into $X$. If the polish space is perfect, one may construct a Cantor scheme on it and as a result there is a copy of Cantor set in every such space.
I don't see a useful infinitary game being played in this construction. In any case, a game without a winning criterion doesn't appear very exciting.
One of the early motivations for considering infinitary games in descriptive set theory was the two way interaction between "nice" sets of reals and the determinacy of the games being played on these sets. Suppose $X$ is a set of reals. Define a two player game $G_X$ on $X$ as follows: player 1 and 2 successively choose either $0$ or $1$ to "construct" in $\omega$ steps a real. If this real is in X, player 1 wins otherwise player 2 wins. A winning strategy for a player is a map on finite strings of $0, 1$ that suggests the next digit to be played and when followed results in a win for the corresponding player. A game is determined if there is a winning strategy for one of the players. The following theorems give a sample of such results:
(1) If $X$ is Borel, $G_X$ is determined; analytic games are determined iff sharps exist.
(2) For a pointclass $\Gamma$ determinacy of sets in $\Gamma$ implies that all sets in $\Gamma$ are Lebesgue measurable and have the perfect set property and Baire property.
Determinacy has deep connections with the theory of large cardinals but I am not competent enough to describe it. The following is a landmark result:
(4) "Large" cardinals imply that all projective games are determined and therefore also that every projective set is Lebesgue measurable and has the perfect set property and Baire property.
