The Adventures of Tow Sawyer was written in 1877. Let's take a look at other familiar works from the next ten years:

• Off On a Comet, Jules Verne, 1877: Science-fiction story of a chunk of Earth that gets struck off by a comet, real world political and racial problems examined in an exotic, isolated setting.

• Anna Karenina, Leo Tolstoy, 1877(finished serial): Realist tragedy about Russian aristocrats and socialites

• The Brothers Karamzov, Fyodor Dostoyevsky, 1880: Philosophical tale with a modernly "meta" narrator

• Treasure Island, R.L. Stevenson, 1883: Exactly what we know it as: stowaways, buried treasures, peg legs, and the trademark pirate moral ambiguity

• Flatland, Edwin Abbott Abbott, 1884: Scathing critiques of Victorian England set in the mathematical equivalent of sci-fi

• Jo's Boys, Louisa May Alcott, 1886: Sequel to Little Women/Men in the same tradition: family drama, romance, and feminism

• A Study in Scarlett, A.C. Doyle, 1887: Sherlock Holmes' debut, with an odd interlude of Romanticism

I hope this gives a clear picture that, across the world, there was a whole variety of literature at this point. Additionally, we can see the foundations of the genres we now love in books, movies, and television. I think you're not only underestimating the diversity of literature, but also overestimating its age. The literature of this period is not many degrees removed from our own, and much of it is still in the popular eye. We're still reading the best works from the period, which was only 130 years ago. You can hardly read about Long John Silver without thinking of Captain Jack Sparrow; where would Bradbury and Assimov be without the pioneering work of Verne? The works are still fresh in our cultural memory. Any modern humorist or novelist has read Twain. Twain's a wonderful author, and he will influence the new works. The real situation is not that Twain is particularly modern, but that modern authors particularly like Twain and take after him.

Twain is sometimes known as America's first celebrity because he was hilarious and relatable and possessed a uniquely American sensibility, plus he was distinctive looking and often made money travelling around the country on speaking tours. He is also extremely quotable and known for his Churchillian witticisms. So a big part of your question goes down to "Why was Mark Twain such a cool guy?" and that is probably impossible to nail down.

However, Twain had a great moral legacy of struggling with race, which is something that is still extremely relevant. When Twain was a child his parents owned slaves, he grew up in a small and quiet southern slaveholding town, full of "good people" who also owned slaves. He matured and began to make a name for himself and began to question a lot of the aspects of Southern culture, and also married into an abolitionist family, all as America fought its bloodiest war ever, went through reconstruction, and struggled to move on and find an identity that wasn't quite so damning. Race is still huge, as are the impulses to look away from it, and it was Twain's best topic, so Twain is relatable partially because he wrote about things we can still relate to.

As Twain got older he began to explicitly criticize the institutions of slavery and mock the South for its backwards ways, and this is often the side of Twain that we are exposed to in school. For example, many people have read his short satirical editorial on lynching, Only a Nigger.

But mistakes will happen, even in the best regulated and most high toned mobs, and surely there is no good reason why southern gentlemen should worry themselves with useless regrets, so long as only an innocent "nigger" is hanged, or toasted or knouted to death, now and then. What if the blunder of lynching the wrong man does happen once in four or five cases!

This is typical of his writing on race after the civil war, and there are similar passages in The Adventures of Huckleberry Finn and Life on the Mississippi. However one merely needs to scratch the surface of moral outrage to find a much more conflicted picture. For example, Twain wrote a sort of confessional essay about his brief stint in the Confederate Army. Twain portrays it as a bumbling comical affair, in which he joins the militia in the heat of secession, gets lost in the woods a lot, tries to avoid any fighting, and then sits out the rest of the war. The introduction to his essay is a good primer on Twain's more conflicted feelings on slavery and southern culture.

Out West there was a good deal of confusion in men's minds during the first months of the great trouble--a good deal of unsettledness, of leaning first this way, then that, then the other way. It was hard for us to get our bearings. I call to mind an instance of this. I was piloting on the Mississippi when the news came that South Carolina had gone out of the Union on December 20, 1860. My pilot-mate was a New Yorker. He was strong for the Union; so was I. But he would not listen to me with any patience; my loyalty was smirched, to his eye, because my father had owned slaves. I said, in palliation of this dark fact, that I had heard my father say, some years before he died, that slavery was a great wrong, and that he would free the solitary Negro he then owned if he could think it right to give away the property of the family when he was so straitened in means. My mate retorted that a mere impulse was nothing--anybody could pretend to a good impulse; and went on decrying my Unionism and libelling my ancestry. A month later the secession atmosphere had considerably thickened on the Lower Mississippi, and I became a rebel; so did he. We were together in New Orleans, January 26, when Louisiana went out of the Union. He did his full share of the rebel shouting, but was bitterly opposed to letting me do mine. He said that I came of bad stock--of a father who had been willing to set slaves free. In the following summer he was piloting a Federal gun-boat and shouting for the Union again, and I was in the Confederate army. I held his note for some borrowed money. He was one of the most upright men I ever knew; but he repudiated that note without hesitation, because I was a rebel, and the son of a man who had owned slaves.

The confusion about the issue evident in this introduction is most famously explored in The Adventures of Huckleberry Finn, Twain's greatest work and his legacy. A close reading of Huck Finn quickly shows that Twain had the tiger of race by the tail. Jim is first used as the butt of a minstrel joke. He thinks he's been rode by witches after Tom messes with him when he's asleep.

Jim was monstrous proud about it, and he got so he wouldn't hardly notice the other niggers. Niggers would come miles to hear Jim tell about it, and he was more looked up to than any nigger in that country. Strange niggers would stand with their mouths open and look him all over, same as if he was a wonder. ... Jim was most ruined for a servant, because he got stuck up on account of having seen the devil and been rode by witches.

But alone with Huck Jim shines forth as the kindest (and only honest) person in the novel. He gets angry when Huck plays a trick on him. He talks about missing his children. Nevertheless he is constantly being written out of the story, especially when other people are around: forced to hide in a swamp, tied up and pretending to be Huck's escaped slave, painted blue as a "sick Arab." Twain doesn't know how to reconcile the minstrel Jim and the Jim that Huck loves as the closest thing to a father figure. Then there is the infamous "Problem of the ending." Right after Huck proclaims that he will help Jim escape slavery even if he does go to hell, the moral climax of the novel, Twain slaps together one of the most groan-worthy endings ever written - an extended coincidence-heavy comedy wherein Tom (the little shit) comes back, Jim is caught by Aunt Polly, they pretend to bust him out via various humiliations so Tom can fulfill his idea of a proper escape and then it turns out Jim was free all along. Then Jim is paid off with a few bucks and its happily ever after. The ending is also notable for the inclusion of Nat, the "Punkin headed nigger" whose "wool" is tied up in little bunches to keep off witches, and Aunt Polly and Uncle Silas, who own slaves and imprison Jim, but are shown as kind people and, moreover, absurdly inept at imprisoning Jim. Nat obviously gives us a return to minstrel show humor, while Aunt Polly presents a clash between the obvious injustice of slavery ("Alright then, I'll go to hell") and the kindness of folks not unlike Twain's own family.

It is also important to note that Huck at the end of the novel refuses to return to the East; he lights out for the territories, i.e. West, a solution to the problem of race and plot that would neatly anticipate America's own, as seen in the Frontier Myth, the career of T. Roosevelt, Owen Wister. All of that is a bit much to get into in this already long post.

So in summation Twain does a really good job of being confused about race which is just about where we still are. Race has arguably been THE defining issue in this nation's history and Twain works at trying to reconcile the good intentions and kindness of the slaveholders he grew up knowing with the barbarism of the institution. The solution that Twain comes to ends up being the same as America's own: look West!

It also deserves to be mentioned that Twain had the great good sense to sound the death knell of the senescent literary movement of Romanticism (there is a wrecked steamboat in Huck Finn named The Sir Walter Scott) and step in with something very very different which also appealed to an American audience who didn't relate much to lords and swords. Although he wasn't strictly the first to do it, Twain's embrace of idiom was the best and most enthusiastic and it was really something at the time. There weren't a lot of sympathetic novels about ignorant white trash, certainly not written by writers of Twain's caliber and fame. A lot of what has come since is founded on Twain's innovations and especially on Huckleberry Finn, and that is a big part of what Hemingway was talking about when he shot his mouth off about Huck Finn. So another piece of the puzzle is simply that Huck Finn is really really influential, especially stylistically, and a lot of people have imitated Twain in a lot of ways.

What authors are you comparing him to?

I'm currently reading "The Adventures of English" wherein he talks about how Twain really tried to capture the spoken English of his day rather than appeal to some aristocratic English of the past. So that is possibly why it's relatable.