This paper proposes a novel integrated sensing and communication (ISAC) receiver design framework for OTFS (orthogonal time frequency space)-based MIMO (multi-input-multi-output) systems from a unified perspective of variational inference. We first construct a factor graph representation for the OTFS-based MIMO system according to the factorization of the a posteriori probability (APP). This representation establishes a direct probabilistic link between sensing and communication, allowing both functionalities to benefit from their integration. On this basis, we develop a low computational complexity message passing algorithm by minimizing the variational free energy associated with the global APP. In particular, belief propagation, mean field, and expectation maximization algorithms for data detection, channel coefficient estimation, and kinematic parameter sensing are derived, respectively. To reduce the communication overhead for the implementation of ISAC algorithm, we propose a federated learning scheme for distributed kinematic parameter sensing. Specifically, by solving the sensing problem in different fashions, three federated learning modes are devised. Simulation results validate the superior performance of the proposed scheme.