The FAA does not differentiate between drones and other aircraft in the regulations. Their action came due to the stupidity of a few drone operators, some complaints on privacy invasion (both real and imagined), some legit concerns (such as flying over forest fires and interfering with the suppression aircraft), association of drones with the mid-east conflict, flying too close to airports, etc. Presumably enough complaints to legislators and the FAA was told to "do something". The registration requirement applies when flying aircraft and drones of 250 grams (about 8.5 oz) or more, both RC and free-flight. Control-line and kites are presumably exempt as they are under the physical control of the operator.
Not that the FAA has any clue about the hobbyist aspect. My personal interest is in the dawn of model aviation planes, the 1930s and 40s, in balsa and silk. These range from 12 to 16 oz or so, but fall under the rules. Commercial drone operation requires special testing and licensing, applies even to a farmer aerial photographing his fields, a realtor photographing a property, and related. The forums are full of comments, even some speculation of rule tightening. The FAA is effectively killing the RC aircraft hobby, an example of restricting the minority to 'protect' the majority.
The registration applies to the operator and he has to have a copy of the registration number in his possession when flying. The aircraft or drone has to have that number permanently attached in an easy-to-read position. The latter is not particularly different than having your name, address, and phone attached. The general safety rules of the AMA are recommended although the wording can easily be interpreted to be mandatory.
Birds are, so far, exempt.