Modular organization and degree-degree correlations are ubiquitous in the connectivity structure of biological, technological, and social interacting systems. So far most studies have concentrated on unveiling both features in real world networks, but a model that succeeds in generating them simultaneously is needed. We consider a network of interacting phase oscillators, and an adaptation mechanism for the coupling that promotes the connection strengths between those elements that are dynamically correlated. We show that, under these circumstances, the dynamical organization of the oscillators shapes the topology of the graph in such a way that modularity and assortativity features emerge spontaneously and simultaneously. In turn, we prove that such an emergent structure is associated with an asymptotic arrangement of the collective dynamical state of the network into cluster synchronization.