Does anyone know how google play music's recommendation engine works?
Deep neural networks, probably with tens of internal layers, distributed across many many machines. Each user probably receives scores from many deep networks for various applications, and each network is probably automatically scored to determine when its particular model is most helpful. This is the pattern used across the industry. Music companies have the additional complexity of trying to parse sound data. Spotify does this with their Release Radar recommender.

As far as targets go, they probably build models to predict all forms of engagement. You probably get scores for listens, scores for time listening, scores for thumbs up or down, scores for add-to-list, and so on. Ensemble methods work well in practice and Google probably gives you on the order of dozens to hundreds of different machine-learned predictions.