Currently, media organization is possible.
The "problem" with the media library in wordpress is always interesting. Check out the following plugin to solve this problem: WordPress Real Media Library. WP RML creates a virtual folder structure based on its own taxonomy.

This allows you to organize your Wordpress library with good folder access. It is easy to use, just drag and drop your files and transfer them to a specific folder. Filter when inserting media or create a gallery from a folder.
Take your WordPress media library to the next level with folders / categories. Organize thousands of images.
RML (Real Media Library) is one of the most popular Wordpress media players plugins. It is easy to use and allows you to organize thousands of images in folders. It is similar to the wordpress category as in the posts.
Use the mouse (or touch) to drag and drop files. Create, rename, delete, or reorder folders If you want to select an image from the "Select Image" dialog (for example, the displayed image), you can filter when inserting the storage medium. Just install this plugin and it works great with all your image and multimedia files. It also supports a multi-level interface.
If you buy, you get: FREE updates forever and high quality and fast support.
From the product description I can quote. If you want to try the plugin, there is on the plugin page .
Update # 1 (2017-01-27): physical download management
Once upon a time, I started to open this stream , and now there is an extension plugin for the Real Media Library that allows you to physically manage the uploads folder.

Check out this plugin: https://wordpress.org/plugins/physical-custom-upload-folder/
Do you know the wp-content / uploads folder? There, files are stored in folders based on year / month. This can be a very complex and massive process, especially when you are working with an FTP client, such as FileZilla.
Move already downloaded files: this plugin does not allow you to physically move files when moving a file in the Real Media Library, because WordPress uses URLs in different places. It is very difficult to support such a process. So this only works for new downloads.
Physical organization on the server?
(Please read if you are a developer). As a developer, I thought about a decision about this. Does it make sense to organize downloads on the server? Yes, I think. Many people ask to organize it physically. I also think that the process of moving files to the server and updating links to images is very difficult to develop. There are many plugins now that store URLs in their own database tables.
Please review this topic where I explained the problem: https://wordpress.stackexchange.com/questions/226675/physical-organization-of-wordpress-media-library-real-media-library-plugin