How Does This Force Thing Work In Physics?

A spring scale measures tension. It doesn't matter if you're pulling down (with gravity) or up (against gravity) or some other direction, it's only the tension on the spring.

(Like everything in physics it would be more complex than that, since the spring scale will have a tiny amount of friction with the sides of the scale and within the spring itself, but they are all relatively small for what you describe.)

Most game-based physics engines don't provide that number because they simplify that part of physics. Objects have velocities and forces and masses and shapes, but the rare cases tension is actually computed it is then used and immediately discarded.

How you would measure that in a game depends on your physics engine. If you were building your own physics engine you would likely calculate the sum of the forces on each end of the spring scale, then take only the force components aligned with the scale's axis.