To be able to explain what is happening, allow me to go in to a little bit of background as to how Windows Phone handles media files.
When you drag and drop files from your computer on to the phone, while it may appear that you are creating an identical structure on your phone, there is in fact more processing taking place in the background. When you connect your phone to the computer, it is not mounted as simple external storage. It is instead mounted as an MTP device.
When you move a bunch of files on to the phone, for each file, the phone automatically creates a metadata file in a hidden folder.
When you try to view/play the media files on your phone, the phone actually looks for the metadata files that link to the actual media files. The metadata files have no directory structure and hence you see your media appear as one long list.
So the answer to your question is: no, you cannot create your own directory structure for media files. Your best bet is to use Playlists (as suggested by @NeilTurner).