The lack of a Godzilla anime or manga is an interesting enigma... to solve it properly, some research would be needed, but if I had to guess, I would say it might be a structural thing. Giant monsters movies are at their heights from the '50s to the '70s and it's an era when japanese studios are highly specialized. Monsters movies was Toho's thing (Daiei tried to compete with Gamera) and I don't believe they were much involved with television nor animation at the time.
Another possibility is that it is an ontological thing: something that has to do with the very nature of Godzilla and other films/series of its kind. I can't think of any giant monster movie (Kaiju) or costumed squad TV serie (super sentai) that ever became an animated serie or originated in a manga. Even metal heroes shows produced by animation behemoth Toei have no animated version. The cause could be that these shows all belong to a meta genre that the japanese call tokusatsu ('special filming' wikipedia tells me) whose main point is the make-believe craft (note that even in low budget series there is a campy charm to it). The fireworks, the puppetry, the miniatures, the costumes, the make-up, the creative editing... something that in a way could be seen as belonging to a tradition similar to kabuki theater or puppet theater, all things that would be lost if using animation. A counter-argument for that theory is the fact giant robots can be found in manga, anime and tokusatsu!