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There are 4 identical cubes with the same specs - cloth and collision modifiers. The problem is that during the collision the green cube is not affected by the blue one at all.
On the left side: green cube is not affected by the falling blue cube, why?
On the right side: they collide as expected.
Is it possible to make the blue cube interact with the green one (compression)? Naturally, the green cube should compress similarly to the blue one.
You can solve this by joining both cubes into a single object, so that they use a single cloth simulation, and then enabling self-collision on the physics:
I believe that each simulation is its own, independent thing, with the order of the simulations managed by the dependency manager. In your example, the green cube is being evaluated before the blue cube (it could go the other way.) The green cube collision stops the blue cube-- so the blue cube's collision never enters the green cube's space. In the case of the two blue cubes on the right, I believe that the later cube is penetrating the collision of the earlier cube, creating some interaction. By joining the two cubes, we create a single simulation to evaluate both of them, rather than two simulations run in sequence.
When using cloth I had the same issue when recreating your scene. But when using Soft Body it seemed to work. See image below
