Short answer, the job is called "narrative writer" here at Riot. (I was the narrative writer on Pantheon.)
Longer answer. To write for games, my typical recommendation is for people to study screenwriting. It's the form of writing that's closest to a craft (like making a table), and not just art. It requires collaboration, clarity, and planning. It's repeatable and standardized. These are the kind of qualities that can help companies not waste millions of dollars. And if you have screenwriting (TV, movie, comic etc) credits, that helps on the job market too. It's a sign you have more than just ideas. You've built words and stories in the trenches.
No degree is required, though... it's about passion and luck, and talent, but talent only after the others. Passion and luck can carry you past the limitations of talent, if you're willing to bleed into your keyboard. (Does it sound like I'm typing from experience? Because I'm typing from experience.)
Anyway, one final bit of advice. If you want to write games... you should write games. It's easier than ever to build your own projects from the ground up. Easier than ever still means it's really, really, really hard. But getting past how hard it is is the first step. No one will hire you to write if you haven't already written. Put in work at the keyboard, chase every opportunity you can. Then wait for luck.