I was looking around my achievement list and saw that I had earned some achievements that say they are attached to rewards, but I do not have the rewards it says I should have. Do I have to do anything specific to claim the reward?

Specifically, the "Leaving a Good Impression" II and III achievements. For II, it shows a Princely Hatchling as a reward. I have neither the minion shown in my owned minions nor do I have an item anywhere in my inventory that would allow me to summon such a minion (on me or my retainers). I would have earned this quite some time ago (I went on a long hiatus), but I think I would have definitely not vendored or "dropped" such an item.

Same thing with III. The reward is shown as a Partisan Crown. This one I would have earned quite recently (like within the last day or two). I definitely don't have this item anywhere in my armoury chest, my inventory, or on any of my retainers. I could have gotten vendor happy as I did just go through some armoury slots to do some cleaning, but I was checking each piece before vendoring them for glamor value, possible materia assimilation, and market boardability, before vendoring it off, so I should have seen this item and I remember anything about it at all. I definitely don't have it right now, and I do have the achievement though.

I think it's pretty unlikely that I would have lost/overlooked two valuable achievement rewards so I think it could be a possibility that I have to do something specific to get or claim them perhaps? How does one actually obtain items as a reward for an achievement?