Thanks to coleopterist's answer, I'm pretty sure I found the specific variation of the basic fairy tale that I read (likely in an abridged form).
According to the wikipedia section on stories influenced by the fairy tale coleopterist provided:
Stephen Vincent Benét's 1937 short story "The King of Cats" based around this folk story was selected by The Library of America for inclusion in its two-century retrospective of American Fantastic Tales, edited by Peter Straub.
Note that wikipedia appears to have some minor errors: the title is actually "The King of the Cats". The text version I linked also indicates a publication date of 1929, not 1937. This site agrees.
After reading through Benét's story, it clearly matches the details I remembered:
The story starts with several characters discussing the startling fact that Monsieur Tibault, a famous and talented musician from Europe visiting Carnegie Hall, has a tail that he uses to conduct the orchestra.
The name Tibault may actually be a nod to William Shakespeare's Romeo and Juliet, as the character Tybalt is referred to by Mercutio as "King of Cats".
Tommy Brooks was in love with Princess Vivrakanarda, who was described as having some feline elements herself ("She moved with a feline, effortless grace" and " She lived alone and was reputed to be very lazy at least
it was known that she slept during most of the day but at night
she bloomed like a moon-flower and a depth came into her eyes. ").
Tommy notices the Princess acting strangely after hearing about Tibault's tail, and witnesses her strange reaction upon first seeing Tibault's tail:
She was not applauding,
her hands were clenched in her lap, but her whole body was
rigid, rigid as a steel bar, and the blue flowers of her eyes were
bent upon the figure of M. Tibault in a terrible concentration.
The pose of her entire figure was so still and intense that for an
instant Tommy had the lunatic idea that any moment she might
leap from her seat beside him as lightly as a moth, and land, with
no sound, at M. Tibault's side to yes to rub her proud head
against his coat in worship.
Tommy's suspicions increase when he witnesses Tibault accidentally rip his clothes, at which point Tommy sees not the expected skin underneath, but rather a coating of sleek black fur.
He then brings his concerns to his friend Billy Strange, who recalls the story of the original fairy tale from a book. Together, Billy and Tommy come up with the plan for Tommy to tell this story to Mr. Tibault.
Tommy gets his chance at the farewell dinner party for Mr. Tibault. Just prior to the dinner, the hostess confides to Tommy that there will be a surprise announcement of the engagement between Tibault and the Princess during the meal.
Tommy then comes unnoticed across the Princess and Tibault and his suspicions are deepened further:
Tibault was seated in a chair and she was crouched on a stool
at his side, while his hand, softly, smoothly, stroked her brown
hair. Black cat and Siamese kitten. Her face was hidden from
Tommy, but he could see Tibault's face. And he could hear.
They were not talking, but there was a sound between them.
A warm and contented sound like the murmur of giant bees in
a hollow tree a golden, musical rumble, deep-throated, that came
from Tibault's lips and was answered by hersa golden purr.
A desperate Tommy manages to tell his story. Tibault calmly questions Tommy on the details of the story, and then cries "Then I'm the King of the Cats!". There's a flash of light and a cloud of smoke, and when it all clears, Tibault is gone.
The Princess later sails off on a sea voyage due to "shattered nerves" from the surprise, so Tommy does not wind up winning her hand. He instead suspects that the sea voyage was an excuse to go find Tibault.