This work addresses numerical instabilities that can appear when computing the mean stress in linear elasticity and coupled poroelasticity problems discretized with low-order finite elements. The linear elasticity and coupled poroelasticity models are solved using both primal and mixed finite element formulations. Stabilization is obtained by enriching the finite element approximation with an approximation of the Laplacian of displacements. This Laplacian is then evaluated with the Physical Influence Scheme (PIS) by leveraging the underlying governing equation. A key step in the proposed stabilization is the calculation of a parameter h, often computed in the literature as a characteristic length of the element. In this work, we calculate h by solving an optimization problem at the element level. To avoid the high computational cost associated with this procedure, a machine learning model is proposed to predict the optimal h. The benefit of combining PIS with an appropriate computation of h is that the resulting stabilization scheme does not rely on any type of heuristic or user-specified tuning parameter, as often required in other stabilization methods. The results show that the proposed stabilization strategy can effectively remove both saddle-point and Gibbs mean stress oscillations in linear elasticity. We also report, for the first time, that mean stress oscillations can also appear when solving coupled poroelasticity problems, and, differently from pore pressure oscillations (which naturally vanish with time), mean stress instabilities are persistent throughout the whole simulation time, unless deliberately removed. The proposed stabilized mixed formulation is able to remove both pore pressure and mean stress oscillations in coupled poroelasticity problems. Finally, the calculation of h is shown to be critical for the quality of the stabilization, with the machine learning-based approach providing the best compromise between numerical diffusion and accuracy.