Multi-connectivity and reconfigurable intelligent surface (RIS) are two promising technologies to tackle the challenging blockage issue in millimeter wave (mmWave) communications. Multi-connectivity enables a single user to associate with several mmWave base stations (mBSs) simultaneously, while deploying RIS can construct virtual line-of-sight (LoS) links to bypass obstacles. For dense mmWave networks integrated with multiple RISs, the user association problem that significantly affects system performance by screening out effective and robust communication links is actually a hybrid association problem, consisting of mBS-user association and mBS-RIS-user association. However, this vital association issue has hardly been investigated yet. In this paper, we consider the hybrid association under a user-centric architecture to mitigate the blocking effect, and jointly optimize the power allocation problem between multiple associated links to maximize the achievable sum rate of the user. To effectively solve the formulated optimization problem, which is NP-hard, a series of reformulations, relaxations, and decompositions are introduced. Subsequently, employing Lagrangian duality and multiple-ratio fractional programming, a low-complexity algorithm based on alternating iterations is designed to solve the subproblems of mBS-user association, mBS-RIS-user association, and transmit power allocation. Finally, the effectiveness of the proposed user-centric association algorithm is verified by numerical simulations.