The reason people think that's what Godzilla is is because even people who actually watch the films usually come away just remembering the monsters and the fights. The human stories in 99% of Godzilla movies are very forgettable, and frequently have very little to do with Godzilla, which I suppose makes sense since so many Godzilla movies were unrelated movies that Toho just slapped Godzilla into at the last second.
For this reason, the best "Godzilla" movies are the ones that straight up don't have Godzilla in them. Plenty of other Toho-universe films that have monsters that appear in later Godzilla movies are better than most actual Godzilla movies, and it's a shame they don't get attention. Atragon I think is based on a novel, which explains why it has a much better story than most.
Godzilla movies should try harder to at least have human plots that are more intricately tied to what Godzilla is actually doing. I know it isn't actually what the series is, but it's what the series should be. The best part of the 2014 one was the two seconds when Kick-Ass was actually helping Godzilla fight the bad guys, even if only in a minor way, because at least then I had a reason to care what Kick-Ass was doing. I'm sure there are a couple more movies where the humans are more significant to what Godzilla is doing, but usually it's the opposite, where the humans are fighting some aliens or something, and then Godzilla shows up to fight a kaiju the aliens are controlling, but it's really disconnected otherwise.
Final Wars is the best because at least the humans in that are basically the X-Men. It's not like the rest of the Godzilla movies, but at least they're entertaining.