With Android, this limitation does not exists making Google's devices a winner at that level. Did you know that you can manage media files without iTunes? The trick is simply not to rely on the stock apps to do it.
If you do a search in the AppStore, you will find a plethora of apps that can deal with all kinds of media files. Some are free, some will cost just a few bucks. Simply forget about the stock apps and manage your favorite files with something else... As you would do on an Android device!
The magic of VLC is back for iOS devices. It can play many formats (if not all) and with the latest version, you can import movies over wifi or Dropbox.
Sharing files between devices and computers has never been easier using Dropbox. Any kind of files can be accessed and used. It's a universal iCloud! Many apps can access your Dropbox account making it even easier to import documents into other apps.
This is the Swiss Army knife for iOS devices. You can connect to all kinds of servers, download any files, use it as a media player or edit text files. It's the ultimate file manager for a mobile device.
Another great file manager is AirDisk Pro. You can easily import and export files over WIFI making it the perfect tools to share documents with friends.
In the end...
There are many apps having the same features, it's up to you to choose what works for you.