In games like DOORS, once you obtain a badge instead of getting the typical Roblox-made badge notification in the bottom right corner, it is replaced with a custom badge notification like you see in Doors. Thanks!
The only thing I could possibly think of is that DOORS almost has their own type of achievement system, in which they disable the badge notification and instead just have their own UI element for it. They probably also just award the badge normally using their system.