You seem to be under a misapprehension of
what DropBox is.
Some apps have the ability to sign into DropBox to save and download files. Those apps treat dropbox as if it were a kind of folder system. You navigate and use it much the way you would a folder on a computer. But that is only within those apps. DropBox also has it's own app. You can use this app to view and download files from DropBox, and if an app is compatible (and registerers with iOS) you can use the Open In… feature to copy the file to the app. It also lets you copy files to and from the Photo app.
Some few apps do sync their file folders with DropBox (when they are opened or manually). This is entirely up to the developer, and has nothing to do with system wide synchronization. 1Password is an example of this kind of app.
The service that synchronizes Apple's apps, and the few developers who have rewritten their apps to take advantage of it, is iCloud. You can read
about iCloud and what it does here. Don't read more into it than it says.