The name of the Christian go is Yahweh, Jehovah, or Eli, or El, or Elohim, so that god should be treated the same as any other god or goddess. The Idea that The Christian god gets special treatment is simply a matter of religious prejudice.
Or tradition.
It's not whether a god has a name, IMO. It's like Mom and Dad. They have names - Mom, Ma, Mother, or Patricia, Mary, Jane, whatever. But as a form of address, it's capitalized. As a noun, it's not.
I'm going to see Mother.
I'm going to see my mother.
I pray to God.
I pray to the gods of fate.
I have a harder time with the he/He problem.
For instance, say my doubting characters says "If God is real, he's a bastard."
Tradition is to capitalize "he", but since the character isn't a believer, that feels off to me. So I leave it lower case. I keep God capitalized because it's being used as a proper name.
Suppose she says, "If there is a god, he's a bastard."
Here, my character's attitudes will determine whether I capitalize God or not. If she's got doubts but still goes to church, then it's definitely God. If she's a committed atheist who resents religion, then it's definitely god. There are a whole lot of in betweens, and I don't think either choice is wrong.