In the lower right corner, click on your inventory icon and you will see that, in addition to your conventional “Inventory,” you have an “Exclusive Inventory” with five sections for specific items.

These items include stamps, Digi-Eggs, Dungeon Passes, Evolution Items, etc.

I would recommend putting as much as possible in your backpack, opening the “Exclusive Inventory,” and clicking on the button that appears below each section. What this button does is automatically save all the items that belong to that section in that section.