There are references to religion itself but these occur in discussions between Bassett and Uncle Oscar. After Oscar first discovers that Paul is interested in horse racing, Bassett tells him that he has been talking with Paul about horse racing for some time. The narrator notes that Bassett says these things "as if he were speaking of religious matters." This is a simile and it is used to indicate how serious Bassett. is. Many people are dedicated to their religious beliefs and practices. Such a dedicated person is "religious" about their faith. This notion of being religious as serious and dedicated is often used to describe other devotions. For example, someone who "religious" about baseball is very serious about it and follows the game closely.
In one of the following lines, the narrator adds, "Bassett was serious as a church." This is another simile. A church is a serious place. Discussions of life, death, heaven, and the struggle between good and evil occur in a church. These are serious issues. Saying Bassett is as serious as a church means that he is as serious as those who discuss such issues. He is not kidding, being flippant, or joking in any way. If Bassett had not been so serious, Uncle Oscar might not have believed him.
A number of paragraphs later, Bassett is still quite serious. He tells Uncle Oscar "in a secret, religious voice" that Paul's gift of picking winners is "as if he had it from heaven." This is a real religious reference. Bassett's manner has been described as religious in terms of how serious he is. But here, Paul's gift is described as a miracle, as if it's been given to him by God. Bassett means that he can not think of another plausible explanation for how Paul continues to pick winners.