No, I don't see a Shinto allusion there. Speaking of religious ties among ordinary Japanese families, Buddhism is just as likely a source of spirituality as Shintoism. That said, there is no sign this is about anything ritualistic or religious.
In preceding passages the father returns home, morally beat and physically different, so much so the kids don't even recognize him. The mother still does. As a loving and worried wife, first and foremost she wants to know if he is okay, what kind of life did he live every day after they were separated, and how he spent his days.
My understanding is the family is uprooted from California and sent to a concentration camp in Topaz Utah. The family is broken up and separated, the kids and mother coming back home earlier than the father. The dialogue you cite is excerpted from the part where the father is just coming back home with a seeming disconnection that shocks his kids and makes him seem like a stranger to them.
The mother and father are talking about different things but not at cross purposes because they understand each other. The mother is apparently most concerned with her husband's well-being and life in the camp, so her complete sentences would have been: "Did you suffer in the camp?" or "Did you eat well?" I would be greatly surprised that a wife and mother would want to know if her husband prayed every day the day he comes back from the internment camp. Even if she is that concerned with religious formality she would first make sure her husband is okay and then ascribe his well-being to kami-sama (the gods).
The father on the other hand of course is more concerned about his family rather than himself. So he is replying "I thought of you every day." (Very likely in Japanese sentimentality and sensibility. This is something people would say in Japanese.) "Every day I wondered how you were." or in a more American way (they are a Japanese American family after all) "I missed you all every day."