Hi there @Medison, thanks for reaching out to us here!
The Dropbox Badge is a different feature that is used when editing Microsoft Office files locally on your computer. You can check out more info on this here if you'd like.
The icon in your menu bar is designed to be present when the Dropbox app is running on your computer, and lets you know the current sync status of your files with just a glance.
However, it's also how you can access other features of the Dropbox app such as the preferences menu and to check your recently synced files.
There isn't a way to disable this icon from appearing in your menu bar I'm afraid, the only way for this not to appear would indeed be to quit the desktop app and stop syncing your files.
Let me know if this info helps!