At first this achievement seems bugged but actually it just has a bad description. Merely clearing the game does NOT count towards this achievement, despite the clear file saying your Loops have increased.
You have to specifically complete all of the story quests in one playthrough, shown in the Info panel. It doesn't need to be your first playthrough so consider treating the first one as practice and a way to unlock further dungeons and items for sale in later loops. Most of the story quests involve either going to a specific person/location, paying a certain amount of money, or waiting some time for the questline to progress. It's straightforward so specifics are unnecessary and each step of the quest is detailed in the Info panel as you reach it.
When it says to finish your exam, just play the game as normal and save the clear file afterward; the achievement should pop the moment you load that save via New Game Plus in the main menu after the credits.