You can save and export FEATool models created in the GUI to MATLAB m-file scripts ("Save As MATLAB Script..." from the "File" menu), and embed or call it from MATLAB function that uses variables and functions to evaluate and compute geometry dimensions and mesh sizes. See for example:

Parametric study for deflection of a bracket tutorialParametric study for deflection of a bracket MATLAB m-file script
kth wrote

I would like to know please, whereas it is possible to operate FEAT tool script as a function in matlab, with the geometry operations (gobj_block) not set as constant numbers, but set as variables instead, which are fed into the script externally.

Then the FEAT tool script function, should calculate the new grid (with same boundaries etc) and derive the new solution, etc accordingly, for various iterations and with another set of geometry variables each time this function is called.