A system of degenerate drift-diffusion equations for the electron, hole, and oxygen vacancy densities, coupled to the Poisson equation for the electric potential, is analyzed in a three-dimensional bounded domain with mixed Dirichlet-Neumann boundary conditions. The equations model the dynamics of the charge carriers in a memristor device in the high-density regime. Memristors can be seen as nonlinear resistors with memory, mimicking the conductance response of biological synapses. The global existence of weak solutions and the weak-strong uniqueness property is proved. Thanks to the degenerate diffusion, better regularity results compared to linear diffusion can be shown, in particular the boundedness of the solutions.