This is not related to DAVdroid (which only synchronizes the server contacts with the contacts storage), but an issue of your Contacts app, which provides a user interface for the contacts storage.
The stock Android Contacts app behaves like this: If a structured address is known (for instance, because DAVdroid downloaded it from your server), the "formatted address" will be shown and not the single components (street, etc.). However, if this single field is empty and you enter something into it, it will be stored as "street".
If you need to edit address components, you have to use a contacts app that supports this.
Google Contacts (version 2.4 and newer) and some vendor-specific contacts apps supports separate address fields.