So here's what seems to be the info on Mystery Disks, from what I've gathered:

They can be won in tournaments, bought in the shop, or earned on adventures, yes? When used in the shrine, they give a monster of that type, similar to the saucer stones in MR3 (because they are attached to a specific breed type.) For instance, winning the second official tournament nets you a Piroro disk every time, I believe.