I agree that Enya and music within her genre fit this description.
New Age does define her music, as well as any where vocals are used more for ambiance than storytelling.
I would also argue that over the history of Rock music, and its countless subgenres, that vocals are sometimes used more for their instrumental voicing rather than to articulate a story.
I have long contended that the lyrics that Scott Weiland sang for Stone Temple Pilots had little meaning. The lyrical voice in these songs were bold and articulate, no doubt. But the meaning of the words seemed to have little value. At he very least I would offer that the meanings for some of these songs are esoteric, and meant for the small circle of songwriters. What always impressed me about Scott Weiland is not the story he told, but how his voice itself was a distinct and viable instrument among the guitars and drums.
Similarly, Korn and bands like it seem to use a percussive lyrical style. While these songs do seem to be telling a story, none of the stories are much different than others told in other songs; it could be argued that Jonathan Davis only had a few stories to tell. What holds impressive for his music is the cadence and syncopation he uses to counterpoint the percussive instrumentation.
Mike Patton (Faith No More, Mr. Bungle, Fantomas, et al), while an impressive songwriter and storyteller to boot, also uses these cadences throughout each of his projects.