It’s actually a marriage registration form. (婚姻届) The two columns refer to the husband and wife, who have to fill info like nickname, home address, and occupation. (You can see where Marine’s already filled out her name and job.)
Basically, she wants you to apply for the job of “Marine’s husband.”
I’m awful at adding notes on mobile, but if no one else gets to it, I’ll translate the full form later.