The GB only shows up next to assignment types in that dropdown when 2 conditions are met.

When you have chosen that assignment type in the Grade Book Set up.
When you have set your Grade Book up for the year.
Both things have to be true.

If it's set up by marking period, the teacher will not see the GB next to the assignment types in the drowpdown

If you need to change how your gradebook is set up reference How can I change my grade book setup?