First, It depends on your blender version. If you are on 5.0 or newer after you assign the action in the action editor you will need to assign the cube slot in the field next to the action selection in the header. Note: The action slot controls won't show until you have an action set.
Yep. You are running into the new data structure.
Slotted actions.
When you create an action for an object, Blender will set this up for you automatically (basically it functions exactly the same)
However if you want to apply that action to a different object, you will need to set it to use that same slot.
You might think this is silly, but now it means you can have the animation for several objects live inside the same data block.
There is just 1 added step for sharing anim data from one object to another.