Hello! You asked why Mark Twain gives the animals in 'The Celebrated Jumping Frog in Calaveras County' the names of prominent Americans.
The story starts out with Mark Twain telling us why he meets up with Simon Wheeler. His friend wants Twain to ask Wheeler about a boyhood companion named Leonidas W. Smiley. However, Wheeler regales Twain with stories about a particular Jim Smiley instead, which bores Twain nearly to death. This convinces Twain that Leonidas W. Smiley is just a figment of the imagination and doesn't really exist.
Wheeler tells Twain that Jim Smiley is just the sort of man who would bet on anything. He says that Jim Smiley had a bull-pup by the name of Andrew Jackson who only ever lost one fight. The bull-pup would let his other doggie competitors bite him and throw him over until the bets multiplied. Once the bets reached a certain high level, Andrew Jackson would then grab the other dog's hind legs and hold on for dear life until the other dog gave up the fight. Twain knew that his readers would relate to the name 'Andrew Jackson' (the seventh President of the United States), as the President himself was well known for his aggressive, bold and tenacious personality. Giving the President's name to the bull-dog also lent a touch of humor to Twain's story.
Wheeler also tells Twain that Jim Smiley once trained a frog to jump, and claimed that it became the best jumping frog in all of Calaveras County. He then bet a stranger on this claim. However, when Jim Smiley left to find a frog for the stranger to use in their bet, the stranger put some quail shot in Daniel Webster's mouth. The frog is so weighted down after this that he loses the bet for Jim Smiley. Daniel Webster was a US lawyer, congressman, senator and US Secretary of State and thus Twain knew that his readers would be familiar with the name. It is interesting and humorous that Twain has the stranger fill the frog's mouth with quail shot: the statesman, Daniel Webster, was well-known for his powerful oratory.
Hope this helps. Thanks for the question!