I believe that Mark Twain is using Pap as a symbol or representation of the attitudes that he (Twain) dislikes. He is using Pap to comment on the kinds of ideals that supported slavery and white supremacy.
In this speech, Pap talks a great deal about how much he hates a country where a man can't have control over his son and a country where a black man has the right to vote and be respected. I think that Twain is saying that people who would believe in slavery and white supremacy are as short-sighted as someone who would believe (like Pap does) that he has the right to do whatever he wants to his son (including take all his money).
So I think Twain is criticizing this whole set of ideas that says that certain people (parents, white people) should have unlimited power over their "inferiors."
Twain comments on so many things it's almost impossible to keep all of his commentary straight. Pap's drunken tirade is of course convenient for the story. It makes him look more foolish and silly and it adds to the message that the American government at the time did not follow reasonableness and equity if it insisted on subjugating one race to another.
It is another point in the book that Twain harshly criticizes society, but if you notice in the same speech, how likely would it be for Pap, in his situation that he finds himself in, how likely is it that he would also run into a black professor that is at the end of Pap's drunken tirade?
His soliloquy, if it can be called that, criticizes the one institution that was on his side. It was the law that kept him from losing Huck, and all the judges in the story were portrayed as gullible, foolish people. The government was also foolish to have abided by the concept of slavery.