Is very simple, although the term "chaser" may not be as commonly used as it was when John Collier wrote his story. A chaser is a drink that is taken to follow the first alcoholic drink.
#1) How are they similar to "the slaves of political parties"?
Answer: Slaves wanted to be trusted by their overseers because this often would grant them privileges that other slaves did not have.
#2) Why does Douglass make this comparison?
Answer: Slaves would seek with great care to please their overseers to gain their trust, just as office-seekers in the political parties seek to please and deceive the people.
I hope it helps, Regards.