Electroporoelasticity model, which is coupled by Maxwell's equations and Biot's equations, plays an important role in water conservancy exploration, earthquake early warning and many other fields. This work is devoted to investigating its well-posedness and analyzing the error estimates for its splitting finite element approximation. We first provide a novel definition of solution which is consistent with the framework of finite element method. Then, we prove the uniqueness and existence of such solution by Galerkin method and derive a priori estimates of high order regularity. Based on splitting technique, we define a splitting approximate solution and analyze its convergence order. Next, we apply Nedelec's curl conforming finite elements, Lagrange elements and backward Euler method to construct a fully discretized scheme. We prove the stability of the splitting numerical solution and the convergence order of error estimates for different unknowns in both temporal and spatial variables. Finally, we present numerical experiments to validate the theoretical analysis. The numerical results also show that our method reduces much computational complexity while keeping the same accuracy compared to classical finite element method.