Hillary Clinton is a robotic, humanoid life form that changes her colors to help her against whomever she battles. She is Mega Man. Perhaps if she doesn't win this election, she can campaign again in the year 20XX.
Bernie Sanders started at the bottom, and through grass roots campaigning and a lot of hard work, made it to the major leagues, ready to duke it out with political heavyweights. He's the little guy, fighting for all the working people and everyone constantly doubts his odds in this fight. He is Little Mac from Punch-Out!!
Jeb would be the third iteration of someone with his last name in the White House. I won't beat around the Bush: he is Mudkip from Pokémon. This is a third generation character that a lot of people still enjoy, but by and large, everyone by this point is over these games.
Martin O'Malley proved at the first Democratic debate that he is a team player, looking out for the best interest of his party and not wanting to hog the spotlight. He's positioning himself to be the Vice President. He is notable sidekick Luigi.
Young Republican upstart Marco Rubio has sped his way up the polls in a hurry. He appeals to younger voters, as he doesn't look like the traditional GOP candidates. He is a Republican with "blast processing," making him Sonic the Hedgehog.
Though he recently dropped out of the race, Democratic candidate Lincoln Chafee used to be a Republican. Chafee is fluid in his beliefs, unlike many politicians, willing to gather evidence and switch allegiances based on what makes the most sense to him. He is Solid Snake, minus the overly convoluted and near indecipherable backstory.
Ted Cruz is a frequently mocked Tea Party Republican candidate. He seems to (poorly) copy his moves from more popular characters around him. He is Dan from the Street Fighter series. Now if he'd only grow that sweet ponytail.
During the circus the Republican debates often devolve into, Ohio governor John Kasich stands out as the most levelheaded and calm. He's able to keep his cool and remain moderate despite all the craziness and people losing their heads around him. As a result, compared to the other candidates, he doesn't stand out. He has remained mostly faceless, like Master Chief from the Halo series.
Most people haven't heard of Lawrence Lessig. He is a Harvard professor running on a campaign of being a referendum president, meaning he will cede his position to his vice president once he fixes what he thinks is the biggest issue our country faces: corporate money in campaigns. This unknown hero just wants to save his land, then return to his normal life. He is Link, hero of the Legend of Zelda franchise.