Grid topology and line parameters are essential for grid operation and planning, which may be missing or inaccurate in distribution grids. Existing data-driven approaches for recovering such information usually suffer from ignoring (1) input measurement errors and (2) possible state changes among historical measurements. While using the errors-in-variables model and letting the parameter and topology estimation interact with each other (PaToPa) can address input and output measurement error modeling, it only works when all measurements are from a single system state. To solve the two challenges simultaneously, we propose the “PaToPaEM” framework for joint line parameter and topology estimation with historical measurements from different unknown states. We improve the static framework that only works when measurements are from one single state, by further treating state changes in historical measurements as an unobserved latent variable. We then systematically analyze the new mathematical modeling, decouple the optimization problem, and incorporate the expectation-maximization (EM) algorithm to recover different hidden states in measurements. Combining these, the “PaToPaEM” framework enables joint topology and line parameter estimation using noisy measurements from multiple system states. It lays a solid foundation for data-driven system identification in distribution grids. Superior numerical results validate the practicability of the PaToPaEM framework.