Those ethnicities are specific to Faerûn; other settings have their own ethnicities.
As you correctly surmise, the Player's Handbook clearly defines the nine major ethnic groups as specific to Faerûn, that is to say the Forgotten Realms, while the Sword Coast Adventurer's Guide contains further information specific to that same setting, given that the Sword Coast is a place specific to Faerûn. The Imaskari, for example, are descended from the people of the empire of Imaskar.
While officially no one setting is the "core" setting for D&D 5th edition, in practice, the Forgotten Realms is given focus in the Player's Handbook, considering that its human ethnic groups are listed there.
However, other D&D settings have their own ethnicities.
For example, Greyhawk. The prevailing ethnic groups are the pale-skinned Suel, the golden-skinned Baklunish (who destroyed each other's empires in an ancient magical cataclysm), the Flan (natives to the Flanaess, the primary continent of the World of Greyhawk setting), the Oeridians (founders of the Great Kingdom), the dark-skinned Olman, and the river-dwelling Rhenee. None of the Forgotten Realms ethnicities appear in Greyhawk, although someone can canonically travel between those two worlds.
In Eberron, humans have varying skin, eye and hair colour, but the setting glosses over the exact details, and does not name specific ethnic subgroups. According to a comment by Eberron creator Keith Baker:
This is a case where canon Eberron simply doesn’t make an effort to accurately model demographics in our world. The premise is that your human character can look like what you want it to look like, and we aren’t concretely mapping skin color to region; essentially we are looking at HUMANITY as a “race” and cosmetic variation within humanity as a player choice. It’s not realistic, and within your campaign you can certainly decide to do otherwise, but it’s not something that will be defined in canon.
Dragonlance has its own ethnic groups detailed here including the Abanasinians, Arktos, Cobar, Ergothians, Horselords, Ice People, Istarians, Kazar, Kharolish, Lemishite, Nerakan, Nomadic Humans, Nomads of Khur, Nordmen, Schallsea Folk, Solamnics, Tarmak, Tarsian, Thenolite, Uigan, and Wemitowuk.
Eberron and Faerûn might be connected, possibly
It is the default of D&D campaign settings that they are connected to one another through the “multiverse,” which refers to all the other planes beyond the Material one the campaign setting is set on. In AD&D 2e, those planes themselves became their own major campaign setting, Planescape. And it is possible for Eberron to be connected to Planescape’s “Great Wheel” of planes, but it certainly is awkward.
Originally, in 3rd edition, Eberron’s cosmology was unique. It had Astral, Ethereal, and Shadow planes, or approximations of them (i.e. spells relying on them still work), but they didn’t seem to connect to the wider Great Wheel or serve the same “connective” value. The planes it does have weren’t other dimensions, but literally moons you could see from the surface of Eberron. And they didn’t neatly line up with the planes of the Great Wheel. And the afterlife and gods were all different, too.
But then, nothing said Eberron didn’t connect to the wider multiverse, per se. Any such connection was left out, but not ruled out. You could definitely make it work—noted expert afroakuma does so in his Planescape Q&As, for example. For that matter, as Quadratic Wizard points out, Player’s Handbook lists Eberron as being one of the campaign settings connected to the multiverse, though it doesn’t bother to explain how.1 The moons seen in the sky could just be lenses into the outer planes, appearing as bubbles in the night sky, for instance. This takes some hand-waving since several of those planes aren’t really any one plane in the Great Wheel, but it can be made to work. We’ll assume connections exists, then.
Those connections, if they exist, are rarely used, but we have Merrix
Even if those connections exist, they are rarely used; Toril doesn’t know about Eberron and Eberron certainly doesn’t know about Toril or the rest of the planes. If people did, they’d almost-certainly seek them out rather than the fairly-extreme approaches to avoiding Dolurrh that some in Eberron take—like joining the Blood of Vol or a cult to the Keeper. We’d hear about it, and by-and-large we do not.
But those connections could be rare and difficult to use. The “lensing” effect that produces Eberron’s moons can also make those sections of the planes difficult to leave for the wider Great Wheel. Outsiders native to them could have restrictions on their behavior that prevents them from explaining the situation or allowing Eberron’s peoples out—or the wider planes’ peoples in.
And then the limited ways out of Eberron would be known to extremely few. Some dragons, presumably. Maybe Erandis d’Vol herself. Tira Miron, at least if she’s everything she’s purported to be, probably would, and so would Oalian. The Aereni Undying Court, perhaps.
But if anyone knows it, then we can be damn sure that Merrix d’Cannith knows it. Because knowing all the secrets and darkest lore about the metaphysics of his world is pretty much Merrix d’Cannith’s raison d’être.
That leads us to an easy explanation: the only person known to still be creating (normal, functional, playable) warforged is Merrix d’Cannith, operating out of an illegal creation forge buried deep within Cannith South’s holdings in Sharn. He is known (to players and DMs, if not to any of the Eberron public) to continue producing warforged for the sake of various experiments. And if he knew about connections to Faerûn? It would be weird if he didn’t send some through.
As for how, Eberron’s always got eldritch machines, which are literally just “plot hook” in industrial form. I prefer that over an awkward trek through Eberron’s moon–lenses to get to the Great Wheel and then somehow get to Toril, because that’s a rather-unlikely journey for a 1st-level character. Merrix d’Cannith has already known to have performed several experiments on eldritch machines, so this is no great stretch.
Faerûn could reasonably produce warforged too
Faerûn is just chock-full of preposterously-powerful mages, and things like golems are already things there. And given Faerûn’s propensity for mages who sit in towers doing research without ever interacting with the rest of the world, for one of those to have pursued the idea of imbuing an artificial construct with real life, succeed, and then not tell anybody about it, is entirely in keeping with the campaign setting as a whole. In comments, nitsua60 mentions that the new Dragon Heist adventure includes new constructs, and in his answer, Matt Vincent mentions the techsmiths of Gond and their Gondsmen, which are much of the way to warforged already. So for there to be just one, or a handful, of warforged in Faerûn is quite plausible.
The key thing that separates Faerûn and Eberron here isn’t the prevalence of magic, but Eberron’s industrialization of magic, which Faerûn hasn’t done. As long as warforged aren’t disrupting Faerûnian societies through large populations demanding rights and recognition, and having disruptive economic effects, then the setting of Faerûn won’t be threatened by their existence.
A semi-aside: connecting Eberron to the Great Wheel
As described above, you can connect Eberron to the Great Wheel, but it is awkward. Below, I’ll describe some of what has to be dealt with in general for this, but since you’re not playing in Eberron there isn’t really any reason you need to pin any of these answers down. Faerûn doesn’t really care about how Eberron links up to its cosmology, just whether or not it does. For the purposes of a Faerûn character, it’s enough to just say “it does” and leave the details of how that works out and how you explain Eberron’s unique features unexplained (and possibly to be explored later, in play, since after all the characters themselves probably wouldn’t know).
Dealing with moons: Shavarath as an example
For example, Shavarath is basically the Blood War: celestials vs. demons vs. devils. In the Great Wheel, the Blood War plays out across the three planes between Hell and the Abyss, with the celestials coming from yet another set of planes, and numerous schemes and flanking maneuvers spreading out across all the planes. In Eberron, that all is confined to just Shavarath.
To make that work, you have to imagine that Shavarath is connected to some particular plane or portion of a plane in the Great Wheel—perhaps, the place where the fighting of the Blood War is thickest. And then there are portals, perhaps, in each of the encampments for the three sides, connecting further on to the Heavens, Hells, or Abyss, making those seem like part of Shavarath from an Eberron perspective.
Afterlife: what is Dolurrh?
The other major issue is the afterlife. If Eberron’s beliefs are true (and while there’s wiggle room to make them not true, the consistency of these beliefs across many cultures and religions makes that a bit of a stretch), souls mostly don’t go to some befitting afterlife, as they do in the rest of the multiverse (and there is not much wiggle room on that!). Instead, they go to Dolurrh, a gray blandness where they are broken down and recycled. The Church of the Silver Flame believes that this fate can be avoided by joining the Flame in death, and some cultists of the Keeper believe that proper devotion to him and/or the Dark Six can result in one’s soul being “kept” in some other (better?) afterlife.
The beliefs of the devotees to the Flame or Keeper fit within the Great Wheel pretty well, and the lack of Faerûn’s particular idiosyncracies (Wall of the Faithless etc.) is no big deal because that is known to be something specific to the Realms set up by Ao, the Realms’ overgod. No overgod is known for Eberron, but presumably it would have one if it is just another Material Plane within the Great Wheel—so it is possible that the unnamed overgod of Eberron has decreed that souls (usually) go to Dolurrh—wherever that is—on death, instead of their usual resting place with the appropriate plane or divine realm. Dolurrh itself might be just a demiplane, for example, since it doesn’t have anything in it and there’s no particular reason it would need to be large.
About gods in general
Talking about gods brings us to another issue: the gods themselves. Gods in Eberron work very differently from everywhere else, being so distant and aloof that you can’t actually be sure they exist at all.
This is pretty easily explained, though: just as Ao requires the gods of the Realms to deal special hoops to jump through (necessitating the Wall of the Faithless), the unnamed overgod of Eberron requires gods there to be extremely hands-off, or limits how powerful the gods there are allowed to be, or bans them entirely.
Regardless, those gods, if they exist, are only those willing to put up with the requirements—which isn’t the major powers of the Great Wheel. Important deities in Eberron wouldn’t be worth even noticing elsewhere; they’re just the only ones willing to deal with the rules. Or quite simply none are (or the rules are simply “no gods”), and the faiths of Eberron are entirely in their believers’ imaginations.