If one were to follow this story to its logical conclusion, then Kratos, a now long retired god, living in a world alien to him, would pass on his skills as a warrior to his adopted child (no way is that kid biologically related to him), and pass away. Likely in sacrifice, or a subtle more humbling way. Passing on the torch to a new god of war.
I think they should have been a bit more creative with the name, maybe something corny like "The God of War". That could have been a bit subtle, and maybe even thematically relevant.