Jack Skellington is the protagonist from Tim Burton's Nightmare Before Christmas, an immortal skeleton and the "patron spirit of Halloween" (according to several wikis). Skeletons are decomposed human cadavers. While in many fantasy, fiction, and folklore stories, creatures of any kind can be extant on their own accord, or even magically conjured into existence, skeletons, like ghosts, are virtually always implied to be tied to prior living humans (or the corresponding animal type that matches their skeletal structure or ghostly ethereal spirit).
Jack's purpose in Halloween Town is to oversee Halloween: to scare people on that holiday in good fun. Like other mythical creatures, he isn't going to be personally known in any real place on Earth. Even if he personally goes out to scare people, it's unlikely he would gain renown that way; he isn't haunting a location, he's doing scares (i.e. "jumpscares") to observe a specific holiday.
So what does all this mean?
Jack tells us in Jack's Lament that "to a guy in Kentucky, I'm Mr. Unlucky, and I'm known throughout England and France."
This is a peek into Jack's human existence. Jack's Lament is a deeply personal song which gives the audience insight into Jack's persona, mood, prowess as a frightening spirit, but also his personality, his preferences, his vulnerabilities.
And so he's literally sharing with the audience an insight into where he died. "To a guy in Kentucky, I'm Mr Unlucky–" why would a skeleton be "Mr. Unlucky?" The most obvious answer would be because it died. Jack Skellington died in Kentucky; or, if you would like to be very specific, the human who died and transformed into Jack died in Kentucky.
Now you might say "well he could have just been traveling through or visiting Kentucky when he died." And that's true. But Kentucky is not an especially populous state nor does it have any major ports, being landlocked. And so if Jack was indeed "known throughout England and France" is it more likely he was a relatively successful famous socialite or celebrity from Kentucky who frequently visited western Europe, or a famous person from somewhere else who was visiting Kentucky and encounted some ill fate?
Kentucky being home to lots of Scots-Irish settlers and having limestone caverns, indigenous sites, and Appalachian mountains with coal deposits means Kentucky has a fairly rich amount of folklore. And much of that Scots-Irish folklore includes the precursor to modern Halloween: Samhain. And what better character to join that folklore than the patron spirit of Halloween himself, Jack Skellington?
Edit: So another angle about the narrative significance of the line "to a guy in Kentucky I'm Mr Unlucky" in the context of Jack's Lament:
The song has a pattern of building up Jack's victories and strengths and confident claims about who he is, then contrasting it with his own self-doubt, restlessness, and existential boredom.
"With the wave of my hand, and a well-placed moan
I have swept the very bravest off their feet.
But year after year, It's the same routine
And I grow so weary of the sound of screams
And I Jack, the Pumpkin King
Have grown so tired of the same old thing
This structure repeats throughout the song. These are build-up statements about Jack's identity. And Jack's origins with his death and a brief mention of his prior human life fits within that broader theme. It's Jack, once again on-theme with the song, proclaiming who he is, and where he's been, only to go back to the chorus with his doubts and laments.