I'd really only consider the 1954 Godzilla to be a horror monster. Even then, it's less a horror movie and more a somber, anti-war film. While there are horror aspects, I think we the audience are supposed to feel sorry for Godzilla. He was was just living peacefully until he was horribly scarred and mutated by the bomb. The 54 Godzilla is less a malevolent destroyer and more an animal in pain who lashes out violently against those (humanity) who hurt him.
Shin Godzilla could fall more into the horror category. He's certainly played up to be a more malevolent and destructive force.