I am not an expert in RL. I have been playing Go for some years.
Let's quote from AlphaZero's paper first:
komi, the rules of Go are also invariant to colour transposition; this knowledge is
exploited by representing the board from the perspective of the current player (see
Neural network architecture).
In the Game of Go, the difference between Black and White except the board representation is the komi (the amount of points that Black has to compensate White in the final count for playing first). Except the presence of komi, there should be no difference in strategy under the same position if colours exchanged. In other words, given a state $s$ of black stones and white stones on the board, if the optimal policy of Black playing first is $\pi$, then if colours of stones on the board exchanged and it is White's turn, the optimal policy for White should be the same as $\pi$.
With this in consideration, there are at least 2 advantages of using a network that represents the board in the perspective of Self/Opponent rather than Black/White.
The first is that it prevents the network from the possibility of giving inconsistent strategies under two representations of the same state. Consider a network $f_\theta$ that accepts the board representation in the order of $(B,W)$, and a state $s = (X_t,Y_t)$ in which $X_t$ is a feature map for black stones and $Y_t$ is a feature map for white stones and it is black's turn. Now consider a state $s' = (Y_t,X_t)$ (i.e. colours flipped) and it is white's turn. $s$ and $s'$ are essentially representation of the same state (except Komi which does not affect optimal policy). There could be a possibility that the network $f_\theta$ gives different policies for these two representations. However, if $f_\theta$ accepts the state as $(Self,Opponent)$, the input to the network would be the same (except the komi feature).
Therefore, this representation would significantly reduce number of states represented by the features vector $(X_t,Y_t)$, which would be the second advantage to training the neural network. If we consider that in Go, the same local position could appear in exchanged colour in another position, the network could by this implementation, recognize them as the same position. A decrease in the number of states could mean a significant drop in parameters and power needed from the network.
The same principle of making use of different representations of the same state is followed in AlphaGo's other training implementations as well, such as augmenting its training data to include rotations and reflections of the same board position.
However, in the game of Chess, this would be a different case. For a chess position, if the pieces' colours are exchanged and it becomes opponent's turn, it would be a different state because the positions of the KING and the QUEEN are not the same for the two colours.