We've seen games do the "Kaiju" thing before to different degrees of success. Rampage did it pretty well for an Arcade style beat-em-up (though it was more King Kong sizes then Godzilla sizes). Godzilla has a had a few serviceable fighting games. However, the only way to properly emulate the size of Godzilla is to put the player in the perspective of something smaller. This is why Shadow of the Colossus is so effective. It's not just big thing next to small thing, but how the player perceives that big thing as a small thing.
God of War and the titan encounters are similar. Even with the camera zoomed out to take in the entire scene, because we play as Kratos and know how Kratos works, the impact the titans have is significant. They feel massive. If you ever played any space sims with massive capital ships, the thing that best conveys their size is flying up along side one in a smaller vessel, particularly during a firefight. See also Homeworld (2 does this really well). Zoom up to a strike craft as it's navigating a battle and the Capital ships look and feel massive. This are all emphasised with the little touches, like bass heavy battle sounds next to pew pew sounds from strike craft, as well as battle damage sending out fragments larger then said strike craft.
In order to do this with Godzilla, we must first accept that the player can't BE Godzilla. The concept would work well in a different type of game, but people want to be Godzilla too much.
How would I do it if I could? Make it a game where you are reacting to Godzilla and co's shenanigans. Like an RTS where the objective isn't to win battles but to minimize losses... the game could even evolve like the old shows where you have to support Godzilla or other friendly Kaijus, or even deploy your own Kaiju countermeasures (like Mecha Godzilla). At certain points it might even become like conventional RTS games, where you have to fight rogue military elements or alien invaders or whatever... Godzilla has a lot of shit to work with.
Maybe even a city simulator where you build defenses and countermeasures and the objective is to hold out as long as possible while you manage evacuations. When Zilla and co. invades it switches to a tower defense style game, where your earlier preparations are put to work. Then when the battle is done and the Kaijus leave, you have to restore the city and research new tech to prepare for future encounters. The game could employ a time lapse feature, which would emphasis how long it takes to rebuild after a kaiju disaster.
There are loads of ideas that can work, but Godzilla has always been some variant of a beat-em-up... and it's never been better then its competitors. I'd love to see a Godzilla game that does something other then this.