You cannot have shulker boxes stacked on top of each other and still access the lower shulkerboxes' content.

For other types of chests, you can access them, also if there are other chests on top of them ("behave transparent").
Of course the shulker box is optically a full block, but it IS transparent, see attached pics:
You can open a regular chest if a shulker box is on top
Shulkerbox lets light through itself (e.g. sealantern underneath it in complete darkness => shulkerbox behaves like transparent block and lets light through

I'd like to know if it is works as intended that shulkerboxes cannot be stacked on top of each other AND accessed, unlike other chest types.

I know the animation swirls up and would optically interfere with the shulkerbox above, but personally I'd love it if you could have a storage like that, with several shulker boxes on top of each other, even more considering it is "behaving transparent", as I pointed out.