A vulcan who thinks he is an elf would not think that he's supposed to have telepathy and thus would not miss it, when he first encounters a real person, he might discover telepathy for the first time.
That's so true. His telapathy would have NOTHING to be aware of.. other than perhaps complete absence of personhood from the holograms. I predict the Vulcan would decide he must be a superior being because of this. Don't forget it's not all about ears.. I doubt the Vulcans and the Elves bleed the same color so early on he would have become aware of his difference.
So his answer would be a different kind of elf, though of what kind he knows not. There are a huge variety of creatures on Middle Earth, thus a Star Trek Alien would fit right in. So this Vulcan would set out on a quest to find his people, a good start for an adventure.
Probably when Frodo finally manages to destroy the One Ring, a few months after that the Holodeck Program would end.
There could be many things, but not Lord of the Rings. Copyright law and all that, and the current license holders of LOTR film material are fairly litigious (they wanted to shut down a pub because it was called the Hobbit). I could see a Star Trek series doing a holodeck pastiche of Lord of the Rings, in the same way that Dixon Hill takes his cues from Philip Marlowe and Captain Proton is basically Flash Gordon, but not actually use Tolkein's legendarium.
You could called it Generic Standard Fantasy otherwise known as "Dungeons & Dragons". the typical Generic Standard Fantasy involves an Earthlike planet with elves, dwarves, gnomes, halflings, half-elves, and humans, typical evil races are orcs, goblins, hobgoblins, gnolls, lizardmen, trolls, giants, dragons, and the undead are skeletons, zombies, wraiths, ghosts, vampires, liches, ghasts, ghouls. Typical standard characters are fighters, rangers, wizards, clerics, rogues, paladins, bards, druids, sorcerers. Those are your basic cast of characters that would appear in a Generic Standard Fantasy Holodeck program.