Holographic multiple-input multiple-output (MIMO) systems represent a spatially constrained MIMO architecture with a massive number of antennas with small antenna spacing as a close approximation of a spatially continuous electromagnetic aperture. Accurate channel modeling is essential for realizing the full potential of this technology. In this paper, we investigate the impact of mutual coupling and spatial channel correlation on the estimation precision in holographic MIMO systems, as well as the importance of knowing their characteristics. We demonstrate that neglecting mutual coupling can lead to significant performance degradation for the minimum mean squared error estimator, emphasizing its critical consideration when designing estimation algorithms. Conversely, the least-squares estimator is resilient to mutual coupling but only yields good performance in high signal-to-noise ratio regimes. Our findings provide insights into how to design efficient estimation algorithms in holographic MIMO systems, aiding its practical implementation.