Please use this thread going forward: Furniture is moved to the household inventory on loading | EA Forums - 5038706
I had the same issue after updating the game with the Cottage Living update (though I do not have the Cottage Living pack). After a few frustrating loads of the game (trying without CC, and making sure MOO was on before loading the house) where the majority of my modular cubes (but not all) would go to the family inventory, I tried moving the rest of the cubes to my inventory as well and then selling them all. I then bought new cubes and placed those, and all of them stayed after I saved and reloaded.
I've now tried saving and reloading the game an extra time for good measure, and the cubes stayed in place this time.
Hopefully the bug can be fixed, but in the meantime this is at least working for me as a workaround.
I have no mods, so I know that's not the cause. I tried what you mentioned, came back in and modular shelves still went missing but this time it was different pieces whereas before it was always the same modular pieces that went missing. In addition, the bunkbed with desk below, the chair, PC and anything on the desk always ends up in inventory also. I got tired of that so replaced with regular beds & desks, but really like the modular shelving/cubes. I thought the new patch would take care of it, but it didn't mention modular shelving at all.
