The symbolism represents both:
Jim Smiley’s dog shares a name with the seventh president of the United States: Jackson. He served as president just before Wheeler’s story takes place, in 1829–1837. Twain (and, perhaps, Smiley) likely named the dog after the president to invoke his determination and fierceness in battle. Jackson was nicknamed “Old Hickory” because of his toughness as a commander in the Battle of New Orleans in the War of 1812. Pres. Andrew Jackson also fits into Simon Wheeler’s Western tall tale because of his years as a frontier lawyer, his populist defense of the common man, and his taste for gambling. Jackson was also perceived as a symbol of democratic upward mobility—a man of humble origins who, through talent and tenacity, rose to the top.
Jim Smiley’s frog Dan’l Webster is named after Daniel Webster, a statesman from New England known for his eloquence. Smiley takes great pride in how well he educated Dan’l Webster, which may be a reason he would name his frog after the famously brilliant orator. While Webster served as a Senator and Secretary of State, he lost his bid for the presidency in 1836. Twain may also have chosen the name as a joke because like the statesman, the frog lost.