In the last years, several mathematical models coupling differential algebraic equations and partial differential equations for describing the behavior of electrical circuits have been proposed in the literature. Most of them investigate the properties of coupled systems that include one-dimensional drift-diffusion equations for describing the highly sensitive semi-conducting elements in the circuit. Here, we extend the results to coupled systems with higher dimensional drift-diffusion models that allow a proper modeling of semiconductor devices constituted of regions with different material properties. For stability reasons, we investigate a monolithic simulation approach and consider two common variants of PDE discretizations for semiconductors in the system: besides a finite element method that may be combined with the Scharfetter-Gummel approach, a mixed finite element discretization. The resulting differential algebraic equations share important properties that allow us to show that their index is always less or equal to two and depends only on the circuit's topology. The information about the index enables us to conclude unique solvability of the associated initial value problems as well as to choose appropriate time-stepping methods for them.