NNs are not binary -- they're fundamentally analog, so for classification problems, they produce probabilities that a test case is in each possible class. A binary "X/not-X" test often comes from applying a threshold to the NN's output.

Quoting from the Universal approximation theory's Wikipedia, "neural networks can represent a wide variety of interesting functions." While they may be much better at pattern recognition, it's possible to produce almost anything with one, including Go moves, if you can devise a method to interpret the outputs.