The phrase "political axe" typically refers to someone's biased or partisan viewpoint in politics. It means that a person has a strong personal interest or agenda, which can influence their opinions and actions. Full definition
You start your heading with Atheism, then talk about socialism and communism as if they were somehow related — they are not, but you seem to be trying to grind a religious axe and political axe at the same time (which seems rather common). (religion.blogs.cnn.com)