I couldn't have felt really and satisfactorily fine and proud and set-up over any title except one that should come from the nation itself, the only legitimate source. (8.7)
Hank holds to a very American way of thinking: that only a democratic vote matters, and that the will of the people is more important than what the people in charge have to say.
The very first official thing I did, in my administration—and it was on the very first day of it, too—was to start a patent office; for I knew that a country without a patent office and good patent laws was just a crab, and couldn't travel any way but sideways or backways. (9.1)
Another American idea here: register your inventions so you can make money from them. Everything else comes second. Twain seems to be commenting on our eagerness to invent new things… then exploit those things for all the money we can grab.
I was from Connecticut, whose Constitution declares "that all political power is inherent in the people, and all free governments are founded on their authority and instituted for their benefit; and that they have at all times an undeniable and indefeasible right to alter their form of government in such a manner as they may think expedient." (13.9)
Hank's moral superiority is self-evident to him… and comes straight from the land where he was born. It's interesting that he mentions the state Constitution instead of the U.S. Constitution. Could that be one of the reasons Twain made his hero from Connecticut—because the wording of the Constitution worked for him? (Otherwise, we might all be reading A New Hampshire Yankee in King Arthur's Court.)
I had started a number of these people out—the bravest knights I could get—each sandwiched between bulletin-boards bearing one device or another. (16.2)
Hanks uses American business sense and advertising tactics to help spread his ideas. It's pretty silly putting a sandwich board on a knight, but it gets the job done. Twain seems to be saying that the knight is more foolish than the advertisement… though he may also be suggesting that advertising itself is kind of silly and undignified.
In my own former day—in remote centuries not yet stirring in the womb of time—there were old Englishmen who imagined that they had been born in a free country: a "free" country with the Corporation Act and the Test still in force in it. (16.3)
Here's a specific law from Twain's time to remind us that British = stinky. The Acts were set up in the 17th century as a way of getting Catholics out of public office: elected officials had to renounce transubstantiation (the turning of bread and wine into Jesus's body and blood) and other Catholic beliefs. It's funny that Twain should say that in a book so critical of the Catholic Church, but it also makes it clear that he hates oppression rather than any specific oppressive institution.
Knight-errantry was a doomed institution. The march of civilization was begun. (39.13)
Hank connects the British knights with general barbaric awfulness. Once he defeats them, things will get better... and they do. Until, you know, they don't anymore. What does that mean for the march of civilization?
The next moment the rope sprang taut and yanked Sir Sagramor out of the saddle! (39.7)
A British knight with his sword takes on an American cowboy with his rope… and look who ends up on top.
Upon Arthur's death unlimited suffrage should be introduced, and given to men and women alike. (40.3)
Again, Hank is overturning a British institution—Arthur's reign—in favor of an American constitutional republic. It's also worth noting that women weren't allowed to vote in America when Twain wrote the book. He was a forward thinking man, that Twain: describing the way things should be rather than the way they were at the time.
They used the Round Table for business purposes now. Seats at it were worth—well, you would never believe the figure, so it is no use to state it. (40.7)
Twain repeats the same trick as the last two entries above. A very American stock market centered at Britain's former Round Table? Yikes.
Slavery was dead and gone; all men were equal before the law; taxation had been equalized. (40.2)
These are all American ideals, which Hank is very proud of making come true. It's particularly telling because it takes place in King Arthur's Britain… not just England, but the England of their greatest mythical hero. Twain may be saying that American ideals would make this idealized version of Britain a whole lot better.