Above: Godzilla 2000 gets it right: tiny people up against a mountainous foe. Colossus is the perfect gameplay template
Big G has plenty of games available. Too bad they range from shallowly enjoyable to embarrassingly deplorable. That second link, however, heads to an NES game that had the right idea - Godzilla needs to be the villain, not the hero. You should be in constant awe of his size and indestructible force-of-nature stature. Gameplay should be an uphill struggle that seems impossible at first glance, but becomes plausible as you gradually exploit Zilla's few weaknesses. In other words, just like Shadow of the Colossus.
See, when you take control of Godzilla, he's not impressive. You completely lose the sense of scale. Destroy All Monsters Melee is cool because we get to see all the crazy monsters designs (and set up dream matches between Toho's best), yet you never once respect the immensity of what's happening - monsters are destroying our society. No conventional weapons affect them, our best offense always fails and there's nothing we can do. It's a dire scenario that isn't best represented by a radioactive dinosaur suplexing a giant robot into the Eiffel Tower.
A proper Godzilla game would unfold just like Colossus, with an unsuspecting hero forced to take down a group of nigh-unstoppable beasts right out of his nightmares. Gigan, Ghidora, Biollante, hell do 'em all. And Godzilla would be the final boss, the last stand, the King of the Monsters. Sweet merciful crap, can we play that game now?