Star Trek has always been at the forefront of ethnic and cultural diversity. In TOS, we had a Russian, a Japanese-American, a Scot(t), Americans, etc. which was groundbreaking for the time. In the TNG era, we had French, Japanese, (black) American, Native American, etc.

But the most unbelievable thing about this (especially for 400 years in the future) is that they all seem to be mono-ethnic. This seems pretty dated even by today's standards just 20-30 years later, let alone 20 more generations.

As far as I can recall, all the multi-ethnic characters seem to be hybrids between humans and aliens. And the only other exception I can recall would be the children of Keiko and O'Brien, which is more of a development than a starting point.

Have we ever encountered a notably multi-ethnic character? As in, a character just happens to note that they have genetic ancestry from more than one area of the Earth?

Doctor Bashir

Doctor Bashir has an Arabic surname, and he claims to have an ancestor with an Anglo-Saxon surname:

BASHIR: Watley? That was my great grandmother's name.

— "Trials and Tribble-ations" transcript (DS9 5x06)

It's possible that Bashir is merely joking (about Lieutenant Watley flirting with him) to make O'Brien uncomfortable. Bashir goes on to say:

BASHIR: No one ever met my great grandfather. This could be a predestination paradox. Come on, Chief, surely you took elementary temporal mechanics at the Academy? I could be destined to fall in love with that woman and become my own great grandfather.

Hikaru Sulu was part Japanese, part Filipino. This was probably not the original intention for the character, but it was included as part of Sulu's characterization later on, when it was realized that "Sulu" was a distinctly Filipino name. This is made explicit in the novelization of Star Trek IV: The Voyage Home.

There was a multi-ethnic character as far back as the original pilot. José Tyler, the navigator on Pike's enterprise, was canonically half American and half Brazilian. Warning: ethnic stereotypes quoted below (source: The Making of Star Trek, via Memory Alpha)

The Navigator. José (Joe) Tyler, Boston astronomer father and Brazilian mother, is boyishly handsome, still very much in the process of maturing. An unusual combination, he has inherited his father's mathematical ability. José Tyler, in fact, is a phenomenally brilliant mathematician and space theorist. But he has also inherited his mother's Latin temperament, fights a perpetual and highly personalized battle with his instruments and calculators, suspecting that space – and probably God, too – are engaged in a giant conspiracy to make his professional and personal life as difficult and uncomfortable as possible. Joe (or José, depending on the other party) is young enough to be painfully aware of the historical repute of Latins as lovers – and is in danger of failing this challenge on a cosmic scale.

That episode of TNG (Up The Long Ladder) where they had the space Irish (Bringlodi). The Space Irish and the Posh Clone society (Mariposans) were humans but they had developed different cultures after hundreds of years apart. The Posh Clone people were clones of five surviving colonists, of a mixture of ethnicities.

At the end of the episode, the Space Irish moved to the Posh Clone planet to interbreed and provide genetic diversity. The children of these two camps would have parents of different ethnicities within the same species.