You collect Shinies by defeating enemies and finding them throughout the level. The game wants you to collect 4200, though you will only need 1825 of that amount to buy the Battlesets. You do not have to be carrying all 4200 at the same time, though it has to be on one save file (aka obtained with one owl). After completing the story you should have collected around 2000 or so. After playing all of the bonus mission challenges, you should have gotten the achievement. If not, replay a survival mission.