I see a lot of people wanting to make Godzilla dragon/water, dragon/fire, dragon/poison or dragon/anything else, but really, the closest you can get is pure dragon.
He's shown to be weak to cold on several occasions, from the blizzard in Son of Godzilla, to the freezer missiles used in Vs Destroyah and the Absolute Zero cannon used in Against MechaGodziila. So it's known that he has an ice weakness.
He also has a weakness to Mothra, but not to insects in general (see: his vigorous stomping of Kumonga, Kamakirus, and Battra). There's no doubt in my mind that Mothra is bug/fairy, so it's also known that Godzilla has a weakness to fairy.
Lastly, Godzilla draws strength from radiation, but too much radiation, as seen in Vs Destroyah, will damage and eventually kill him. So we can deduce that Godzilla is both strong against, and weak against a substance that is essentially his own essence. So we'll say he's got a dragon weakness.
He shares none of the other weaknesses seen in water*, fire, or poison types, and the weaknesses he does have are all those that only a pure dragon type would have.
If we look at Haxorus, we can even see a stat distribution that, aside from the weak sp. attack, could adequately describe Godzilla.
However, as an aside, I'd like to add that his breath attack is most likely fighting-typed, if we compare its effectiveness against other kaiju.
*Godzilla is shown to be weak against, neutral to, and empowered by electricity at various points in the Showa series. There is no type combination that is at once weak to, neutral to, and SE against electricity, so this has been ignored.
