If I understand correctly, I think this might work:
Assign one shader to all the objects that you want to change color. In Unity, go to the shader's location in your project directory.
Create a state, or find the state that you're hoping will change the color of the shader when it's activated. Drag the shader from the project window directly into the state (as though you were dragging another action into the state from the action library).
A drop down window will appear, click "set property". A new action will automatically appear in the state, "targeting" the shader. From here, you can access the shader's properties (by clicking the "property" drop down menu), such as color. Doing so will affect all objects assigned with that material, since you're changing the materials itself and not the object.
Hopefully that does the trick!