The Illiad was like the ancient Greek Bible because it was used to provide behavioural guidelines and views common to all Greeks. It was a text central to any Greek's education. How does a leader act? Read the Illiad. How are battles waged? Read the Illiad. What is the relation between humans and gods? Read the Illiad.
Alexander the Great was known to carry a copy of the Illiad with him at all times, and many philosophers routinely used excerpts from it to illustrate their points. And people would reference it like we reference things from the Bible in the west (Judas, turn water into wine, cross to bear, turn the other cheek etc)
I know you know this and likely just mistyped, but for clarity, the Iliad is about the war itself. The Odyssey is about the dude trying to get back home.
They weren't religious texts per se, but they were certainly cultural touchstones which contain important lessons about the human condition, death, love, and what's truly important in life. They would fill a role similar to histories in the Old Testament, probably, like the book of Ruth.