The promising gains of reconfigurable intelligent surface (RIS)-assisted multiple-input multiple-output (MIMO) systems, in terms of extended coverage and enhanced capacity, are critically dependent on the accuracy of the channel state information. However, traditional channel estimation (CE) schemes are not applicable in RIS-assisted MIMO networks, since passive RISs typically lack the signal processing capabilities that are assumed by CE algorithms. This becomes problematic when physical imperfections or electronic impairments affect the RIS due to its exposition to different environmental effects or caused by hardware limitations from the circuitry. While these real-world effects are typically ignored in the literature, in this article we propose efficient CE schemes for RIS-assisted MIMO systems taking different imperfections into account. Specifically, we propose two sets of tensor-based algorithms, based on the parallel factor analysis decomposition schemes. First, assuming a long-term model - where the RIS imperfections, modeled as unknown phase shifts, are static within the channel coherence time - we formulate an iterative alternating least squares (ALS)-based algorithm for the joint estimation of the unknown phase deviations and the communication channels. Then, we develop the short-term imperfection model, which allows both amplitude and phase RIS imperfections to be non-static with respect to the channel coherence time. We propose two iterative ALS-based and closed-form higher-order singular value decomposition-based algorithms for jointly estimating the channels and the unknown impairments. We also investigate the computational complexity and the identifiability of the proposed algorithms and study the effect of various imperfections on the CE quality. Simulation results show the effectiveness of our proposed tensor-based algorithms in terms of estimation accuracy and computational complexity.