Movable antenna (MA)-enabled multiple-input multiple-output (MIMO) systems offer a promising enhancement for integrated sensing and communication (ISAC) applications. Unlike conventional MIMO systems with fixed-position antenna (FPA) arrays, MAs can flexibly adjust their positions within a given region, enabling reconfiguration of both communication and sensing channels with additional spatial degrees of freedom. In this paper, we propose a unified mutual information (MI) framework for MA-enabled MIMO ISAC systems, where MI characterizes communication performance as reliably conveyable information and sensing performance as extractable target information in cluttered environments. We formulate an optimization problem to maximize the weighted sum of communication and sensing MI by jointly optimizing the transmit beamforming matrix under a transmit power constraint and the MA positions under practical constraints, with a weighting coefficient characterizing their trade-off. To tackle the non-convexity arising from the log-det objective, position constraints, and the nonlinear coupling between optimization variables, we develop an alternating optimization-based algorithm that iteratively updates the transmit beamforming matrix and the MA positions. Specifically, with the fixed MA positions, we optimize the beamforming by approximating the objective function using weighted mean square error and majorization-minimization methods, yielding a closed-form solution. Moreover, with fixed beamforming, the MA positions are sequentially refined by decomposing the position optimization into simpler subproblems, resulting in an efficient suboptimal solution. Numerical results show that the unified MI framework with MAs significantly outperforms conventional FPA systems in both communication and sensing. Channel amplitude heatmap visualizations further illustrate how MA positioning strategies exploit spatial flexibility in array geometry to enhance overall system performance.