They come, independently, from two different cultures but have a the same root. That root is helplessness. The commonality is that there are things that are bigger than you are and that you are insignificant in their eyes.
Giant monsters have been a staple of most cultures: dragons (western and eastern), giants, hydra, giant dogs/wolves, etc. They often represent an enemy or natural disaster that was represented as a giant monster in stories to help the listener understand how it felt to face the situation.
HPL was pointing out personal helplessness. No matter what you do, your actions are either part of the creature's plan or, more often, have absolutely no effect on the outcome. In fact even cults that try to summon the Old Ones early fail. Why? Because, when the stars are right, they will come again. Nothing we do can speed that up or slow it down.
Godzilla, maybe the beginning of the modern kaiju market was produced when Japan was still reeling from having their invincible imperial army curb stomped by an upstart nation with no history. It was a way of expressing the helplessness they felt to have things so turned upside down on them. Gozilla used radioactive attacks but they were superfluous, it caused absolute destruction even without it. The breath weapon was just putting wings on the tiger. The main point was that their military was absolutely powerless against it. Everything they tried failed. That was the helplessness that generation was feeling. Now you often see kaiju that represent the environment.