I was wondering if the New Californian Republic is really Christian? Does the NCR have anything to do with Christianity? Well ofcourse it is possible there are christians amongst them, but does the NCR recognise religion an important part of post-apocalyptic life?

In the letter you receive from ambassador Crocker, it ends with: "God Bless the New Californian Republic!"

Perhaps it's only because it was a pre-war custom, and it's associated with the pre-war democratic US?

Congress shall make no law respecting an establishment of religion or free exercise thereof.

If the NCR is modeled on the USA, I am sure that tenet is as highly important in their codified law as it is to the USA.

Why does it have to be the christian god just cause it says "God Bless the New Californian Republic!"?

Could very well just be a piece of the past that people hold onto to give explanation for some things where the concept of god passes down from generation to generation but the religions of the old world isn't very widespread. My guess is that people are mostly just live and let live with agnosticism, atheism and deism cause apart from religious factions the concept of religion don't seem all that important to most people in the wastes.