This really depends a lot on how a clipping mask is created.
You can have layer-level clipping masks which mask everything on that layer. Layer-level clipping masks are created by selecting a path and choosing the
Make Clipping Mask from the Layers Panel Menu. What this does is mask everything on that layer based on the designated shape.
An object-level clipping mask is created by selecting more than one shape and then choosing
Object > Clipping Path > Make. This creates a clip group where only objects within the group are masked by the designated shape. (Using the "draw inside" mode creates object-level clipping masks as well.)
I don't know the under-the-hood specifics of how it technically varies, but I do know it varies. There's no way to tell which you have other than looking at the layers panel:
Clip Group in the object level mask.
You can have as many object-level clipping masks as you desire on any layers. There's basically no limitation.
However, you can have only one layer-level clipping mask per layer.
So in short, you need to construct things with Object-level masks, not layer-level masks.
Note: Simply dragging the clipping path outside a clip group does not create a layer-level mask. And conversely dragging a layer-level clipping mask into a group does not create a clip group. Although logically you would think it should. This is a difference at the point of creation which can not easily be "swapped" when dealing with existing objects.