When gifts are opened after doing tasks in "jobs", where do they end up? They're not in any inventory. They just seem to float away when clicked on.

Yes correct, it's part of the quest! Not a decor item
Take example from party job. Let's say when you start a party job, a window will pop up saying that you need to collect 9 party hats. Once you collect all 9, it means you mastered the job and another pop up window will show up saying you mastered it

Ahh, thanks. makes sense.