Dropbox limits apps to storing files below the app folder. The subfolders below the app folder are not shareable. Due to this limitation, sharing a diagram file via Dropbox is not currently possible.
Even if you move the diagrams.net app folder to a different shareable folder within Dropbox and open a diagram file, Dropbox will automatically move the diagram back into a subfolder under the app folder when you save the diagram.
This is a limitation of the Dropbox app folder architecture: they only allow apps to write to a specific folder and that folder cannot be shared.
At this time, only Google Drive implements file sharing in a reasonable way.