Vtyjkyuk

There are 9 human ethnicities mentioned in the PHB (pages 30-31): Calishite, Chondathan, Damaran, Illuskan, Mulan, Rashemi, Shou, Tethyrian, Turami.

PHB says (under "Human Names and Ethnicities" on page 30):

In the Forgotten Realms, nine human ethnic groups are widely recognized, though over a dozen others are found in more localized areas of Faerûn. These groups, and the typical names of their members, can be used as inspiration no matter which world your human is in.

Then SCAG (pages 100-112) mentions another 10: Arkaiun, Bedine, Ffolk, Gur, Halruaan, Imaskari, Nar, Shaaran, Tuigan, Ulutiun.

SCAG says (under "Human Ethnicities in Faerûn" on page 100):

Nine human ethnicities in Faerûn are detailed in the Player's Handbook. Several other noteworthy groups of humans are discussed here. Some are significant minorities in regions or nations that border the North, while others are prevalent in parts of the world far from the Sword Coast.

The wording of these quotes implies (at least, the way I read it) that all 19 of the listed human ethnicities exist exclusively in the Forgotten Realms (note that I'm equating Faerûn with the Forgotten Realms; this may be an incorrect assumption on my part). I am also under the impression that the Forgotten Realms is the default setting for 5e, hence why the PHB includes such ethnicities and quotes relating to the Forgotten Realms.

However, other settings for 5e exist; Eberron, Greyhawk and Dragonlance, for example (I don't know if there are any more major settings; these are the ones mentioned in the back of the PHB and SCAG, at least). Do any of these human ethnicities also exist in those settings, or are they exclusively a Forgotten Realms thing? And if the latter, I'm curious to know what human ethnicities do exist in these other settings.

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 specic 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.

In Greyhawk, for example, 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.