>>9335058She's the first apprentice. Meaning either the first to follow Death-Sensei or the one who caught his interest the most. She's going to get the throne someday.
Mori entered a Reaper caste, either through birth or through her choice (as her family also did?) She had to compete, take classes, pass tests, etc. to show why she should be elevated above the others. It seems like anyone in the Underworld can become a Reaper if they show the right traits. They do age, they do train - possibly from a young age, and they do eventually retire.