Overall I love the idea and execution of the game mechanics.
Random thoughts I had while playing presented in no particular order ... please ignore anything that does not resonate with your vision for this game.
There was a game way back in the day called Mail Order Monsters.
Gameplay wise, it did not have a strong or compelling loop. I enjoyed stomping around and the way destruction was accomplished, but I did not feel emotionally tied to the story. The idea of collecting something, or getting upgrades would help a lot in this area.
One issue was that the story text disappeared fairly quickly, and I did not get a chance to read all of it. Maybe having it pause until the player clicks a button would help.
I think any general game design ideas would work here. Having a tough enemy that you need to fight a certain way would help with the level of challenge.
The idea of upgrades might seem weird with a giant monster. I could see that it might not work. But it could be done more organically. Maybe by destroying the city's power station you could absorb electrical energy and get a lighting beam for instance, something that could cut buildings in half like Shin Godzilla?
As a fellow developer, I would also suggest focusing on the main loop a bit more before doing things like making new kaijus or enemies. Having two static kaijus will not be much more fun (for me) than having one. But having a monster with a more clear objective and goals along the way, with some kind of progression would be really fantastic.
Another gameplay idea is unlocking areas. This is used in more exploration based games. You would have to tailor it to make sense in this context. It would not be "find the key to unlock the door to the boss room." But maybe "destroy the power generators to disable the electric defense system, so you can reach the downtown area..." Something like that.
As mentioned Movie Monster game was also a good one. They had one game type where the humans had captured your monster's child. If you roared, it would respond, and give you a clue as to where they were hiding it. Something like this would also tie our emotions to the monster, who is now defending its offspring, and not just destroying stuff. (again, I did not get to read the full story)
Anyway, just awesome! I hope you sell a million copies :) I will be following your progress.