There's no standard title for Andersen's tales, since they've been published in a bunch of editions and translations over the years. We're using a translation from the original Danish of his complete tales titled Hans Christian Andersen: The Complete Fairy Tales and Stories, so that's what we're going with.
The titles of the various booklets that Andersen published his tales in can give us a few minor clues as to his intended audience and writerly goals. The first booklet of tales, which appeared in 1837, was titled Fairy Tales, Told for Children. Other booklets boast descriptive titles such as, New Fairy Tales and New Fairy Tales and Stories. These titles show us that he thought of his collections as primarily containing fairy tales, though, really, the genres of his stories ranged in topic and theme. And, at some point, he stopped focusing explicitly on kids as his primary audience… the titles and themes of his work aren't all exactly kindergarten-friendly. There's a ton of death and other grim stuff, you know?
So maybe jazzy titles weren't Andersen's strong suit. But once you crack open his stories, you'll see where all his creativity went.