I’m no expert, but as one of the commenters notes, if they were seeking a copyright on the name “Disney’s Princess Aurora” or something along those lines it would indeed be narrow enough. But “Princess Aurora” alone seems unnecessarily broad and potentially troublesome, especially since Disney plucked name, character, and story out of the public domain in the first place.
I just want to clarify something. Disney has filed a trademark
application on Princess Aurora. That is very different from registering a copyright.
A trademark is a mark which indicates a person/entity's products or services. For example, "Google," "Porsche," "Duncan-Hines" are all word marks. There are also symbol marks such as Mercedes-Benz' star within a circle or Disney's Sleeping Beauty's castle symbol. For performers, a name can also be a mark. For example when Ben & Jerry's released the ice cream flavor "Cherry Garcia," they received a very nice Cease and Desist letter from Mr. Jerry Garcia's attorneys (obviously, this was settled).
Almost all corporations (not just Disney) are insanely protective of their marks for many reasons, among them that if they do not guard their marks it may weaken the mark and they could eventually lose the mark altogether. (For example. if you started a website with a url of ILovePorsches.com just see how long it takes you to get a C&D letter from their trademark counsel).
A copyright is a protection of expressed work. In the United States, registration is not required for a work to be copyrighted (for example, as soon as I am done with this post, it will be considered copyrighted). There may be some overlap between trademark and copyright, but they are not the same thing (for example, drawings of Mickey Mouse would be copyrighted, but "Mickey Mouse" is also a mark of the Disney Corporation.