Most of my rifle and all my PCC cans stay attached the guns they are used with most. I keep the remaining few cans on a small shelf in one of my gun safes (actually the remaining rifle cans are on a shelf in a rifle safe, but pistol cans are stored on a shelf with pistols in a smaller safe).
All the parts, springs, spare pistons, spare mounts, shims, tools, etc, are all stored in a large fishing tackle box with compartments for the small parts. The larger parts are kept in separate labeled ziplock baggies in the main large compartment.
So for example, my modular TiRant stays in short mode mounted on my CZ P-10C using a Griffin EZ Lok mount. The extra section of tube and baffles are in a quart ziplock freezer baggy labeled "TiRant 9" in the tackle box. The Alpha thread adapter that fits it is in a baggy labeled "Alpha parts", and there is another baggy labeled "3-lug adapters" that has an Alpha 3-lug that fits it. In one of the tray compartments are spare EZ Lok mounts that can be used to mount it if I got another pistol or threaded barrel.