MANILA—Two of the Philippines’ most famous names entered the country’s 2016 electoral race, with champion boxer Manny Pacquiao declaring himself a candidate for senator, and the late Philippine dictator Ferdinand Marcos’s son announcing a vice-presidential run.

For both Mr. Pacquiao, a congressman, and Ferdinand Marcos Jr., a senator, the higher offices are a potential springboard to the presidency itself, in a country where celebrities and members of powerful clans dominate the political arena. Their candidacies, both announced...