In combinatorial game theory, the disjunctive sum ($\oplus$) of two games means playing the two games simultaneously where you can move in either one game or the other. We can show that for any impartial game $G$, we have $G \oplus G = 0$, which suggests that the disjunctive sum is similar to the symmetric difference of two sets, since "equal" games just "cancel out".
Since the nim-sum (bitwise xor) is just the symmetric difference between the binary representations of two numbers, is there a sense in which the nim-sum is just the disjunctive sum but at a different "scale", i.e. the disjunctive sum applies over games and the nim-sum applies over their Grundy values / nimbers?
Yes, that is precisely the point about Grundy values: The value of the disjunctive sum of $n$ games is the nim-sum of all the games' Grundy values.