Kalman filtering performance is highly sensitive to model mismatch and noise covariance tuning. Learning-based approaches address these limitations but typically rely on supervised training with large datasets and do not produce consistent uncertainty estimates. In this paper, we propose a self-supervised Hybrid Adaptive Kalman Filter that learns structured corrections to system dynamics and process noise covariance from measurements alone while preserving the probabilistic structure of the filter. This allows the innovation likelihood to be computed and subsequently used for model classification via generalized Bayesian inference. Experimental results on real-world and simulated datasets demonstrate improved estimation accuracy and statistical consistency as well as robust classification performance across both low-data and large-data scenarios.