Multiparameter estimation is a general problem that aims at measuring unknown physical quantities, obtaining high precision in the process. In this context, the adoption of quantum resources promises a substantial boost in achievable performances with respect to the classical case. However, several open problems remain to be addressed in the multiparameter scenario. A crucial requirement is the identification of suitable platforms to develop and experimentally test novel efficient methodologies that can be employed in this general framework. We report the experimental implementation of a reconfigurable integrated multimode interferometer designed for simultaneous estimation of two optical phases. We verify the high-fidelity operation of the implemented device and demonstrate quantum-enhanced performances in two-phase estimation with respect to the best classical case, post-selected to the number of detected coincidences. This device can be employed to test general adaptive multiphase protocols due to its high reconfigurability level, and represents a powerful platform to investigate the multiparameter estimation scenario.