Can one access the global stiffness matrix and elemental stiffness tensors in FEATool Multiphysics?

In order to implement our methods the main limitation is sensitivity analysis. This requires us to have access to the global stiffness matrix and the elemental stiffness tensors for each physics model (thermal conductivity, fluid permeability, Youngs modulus, etc).

Yes, you can both modify and define your own equations and PDEs via the weak finite element formulation, and also assemble and use the corresponding stiffness matrices. This is an advanced feature and the Poisson equation example gives a light introduction:

Poisson equation example using the weak FEM formulation
Poisson equation example using the core FEM matrix assembly functions

For a more advanced example you can see the topology optimization model.