Autonomous underwater vehicles (AUVs) show much promise in environmental sensing, aquaculture, and security applications. Robust and adaptive control strategies can immensely benefit these scenarios by increasing autonomy and endurance. However, AUVs are nonlinear systems whose dynamics are challenging to model, especially during agile maneuvers at high angles of attack. To better capture these nonlinear effects, this paper proposes a physics-informed system identification scheme that combines prior knowledge of the system dynamics with data-driven regression. Strategies including Sparse Identification of Nonlinear Dynamics (SINDy), nonlinear least squares regression, and Gaussian processes (GPs) are used to learn the AUV dynamics online from measured data. These data-driven models are then implemented in an adaptive model predictive controller (MPC) for agile maneuvering that drives the system to a set point while updating the prediction model when new measurements are available. The performance of these three system identification strategies is evaluated on two different 6-DOF AUV platforms. All three strategies show good real-time performance, while the GP model offers the best balance between accuracy, speed and robustness. Field experimental data from the SAM AUV and the MOLA AUV are used for performance evaluation.