From what I understand achievement refers to like a breakthrough in jins physical ability like the fight with his first trainer the sword was an achievement since Jin surpassed his limits and broke the sword hence the sword shard was an achievement at least in my understanding at least
Cause he's not affiliated with any magic tower, and he needs to pass through the test first if he want to join unless the examiner/above people allow him to just join without taking the exam.
Its an insane power gap but the examiner might want to see how other applicants will deal with him too, considering they let becky who have killed other applicants before to join the test again
i'm so scared that the key keeper will betray him ╥﹏╥