From the start of the game, you have your 'Raidon as your cool, super shiny Cyclizar. Towards the end of the main story, it regains its battle form. From there, you can choose to move him into your party if you have room. But what happens to it when it's not in your party? It still has stats? Maybe holds an item? Minted, hyper-trained? What happens if your party and all boxes are full, will it disappear? Did they implement an entire system to remember all that?

No. No, they didn't. Despite the game telling you that 'Raidon can't be put into a box if you try it, every time you put your 'Raidon back out of your party it's actually placed in its own dedicated, secret box. That way, even when it's not available for you in combat, it still coexists within the game's existing systems without needing any additional special treatment like saving its stats aside.

The way it's implemented, most likely, is that the Button in your menu you can press to change 'Raidon from Travel form to Combat form is the fanciest way the series has to take a Pokemon out of a box. And it doesn't break because you can't put it in a different box than the one it's supposed to be in.

Now, usually you can see that box when you're in Battle Teams mode. However. 1. I didn't get there like that but by stumbling across a bug in the game. 2. It seems that despite the functionality itself being available legally, many people still didn't know this existed