You're right that Star Trek species are generally portrayed as monolithic societies1. What makes humans special is that they don't have that. Our Hat is not to have a Hat.
Ambassador Soval reveals to Admiral Forrest that this is the opinion of the Vulcan High Command in Enterprise season 4 episode 7, "The Forge"2:
Soval: [The Vulcans] don't know what to do about Humans. Of all the species we've made contact with, yours is the only one we can't define. You have the arrogance of Andorians, the stubborn pride of Tellerites. One moment you're as driven by your emotions as Klingons, and the next you confound us by suddenly embracing logic.
Forrest: I'm sure those qualities are found in every species.
Soval: Not in such confusing abundance.
So the Vulcan opinion is that non-human Star Trek species are, with very few exceptions, as monocultural as they're portrayed on the show.
An alternative opinion is expressed in "The Cage", the original pilot for TOS3. The premise of the episode is that Captain Pike has been taken captive by the Talosian species, who capture and study members of other species. One of the Talosians, referred to as The Keeper, has this to say regarding humans:
The Keeper: The customs and history of your race show a unique hatred of captivity.
The Keeper: No other specimen has shown your adaptability.
The Talosians have a similar, but slightly different perspective to the Vulcans: they believe that Humans are uniquely disposed against captivity (although it's not clear how many sentient species they've studied), and are uniquely able to adjust how they respond to situations. Your mileage may vary, but as far as I'm concerned this is another way of saying that the human Hat is to not have a Hat.
As far as I know, there's no in-universe explanation for this.
One possibility is that it's because Earth has been a united planet for so short a time; Memory Alpha tells me that the United Earth government was founded in 21504. We've observed in real life that globalization tends to limit cultural diversity, so it's conceivable that a few hundred years of single-government would eventually reduce us to a Planet of Hats.
1 TVTropes refers to this as Planet of Hats
2 I should point out that it's possible this is just Vulcan prejudice; one of the recurring ideas of Enterprise was that Vulcans tend to look down on species they view as inferior, and they view all species as inferior in some regard. So this may not be the most unbiased view, but it does mesh with what Star Trek shows us of non-human species
3 Although "The Cage" was ultimately rejected, pretty much all of the footage was re-used in the later TOS two-parter "The Menagerie", so this is still canon
4 The year before Enterprise begins, about a hundred-ish years before TOS, and about 200 years before TNG and its spin-offs. To contrast, the idea of a unified Vulcan society most likely begins with Surak and the Time of Awakening, sometime around what Earth would consider the year 300