We report on self-organization of adaptive networks, where topology and dynamics evolve in accordance to a competition between homophilic and homeostatic mechanisms, and where links are associated to a vector of weights. Under an appropriate balance between the intra- and inter- layer coupling strengths, we show that a multilayer structure emerges due to the adaptive evolution, resulting in different link weights at each layer, i.e. different components of the weights’ vector. In parallel, synchronized clusters at each layer are formed, which may overlap or not, depending on the values of the coupling strengths. Only when intra- and inter- layer coupling strengths are high enough, all layers reach identical final topologies, collapsing the system into, in fact, a monolayer network. The relationships between such steady state topologies and a set of dynamical network’s properties are discussed.