The sum of two games means a turn consists of playing in either game. Then by the mirror strategy, for any impartial game $G$, $G+G=0$, a second player win. That means that under this sum, equivalence classes of impartial games form an abelian group of exponent $2$, or an $F_2$-vector space. XOR describes the additions with respect to any basis, not just the standard one.

You can change the connection with XOR by changing the sum operation. If in a collection of games, you must play in at least one, but up to $k$ games, then the Sprague-Grundy values of the components matter, but the second-player wins correspond to collections whose sums without carries have all digits divisible by $k+1$. This was [observed by Moore][1] where the games are Nim piles. Of course, this is related to [your other question][2]. There are other generalizations possible, where you are allowed to play in other collections of multiple games, and you don't simply XOR the values of the components. 


  [1]: http://www.jstor.org/stable/1967321?seq=1#page_scan_tab_contents
  [2]: http://mathoverflow.net/questions/209933/generalization-of-sprague-grundy-theorem