So, in case anyone was interested about what's going on with the C gods at the moment:
If I were to guess, Special Ami is saying "frog boy" reminds her of the title "Rain King", more so than the event. Sensei may be intended to be the Rain King but has been acting like a Frog boy instead.
Pareidolia also recently brought up a "mother frog":
And may actually just be using this Special Ami in Yumi's Chapter 4 'Frog boy' event, as a cover, considering she suddenly talks in lower case:
Pareidolia might be using Ami's image, due to Sensei trusting and loving her, which makes it easier to manipulate him, through her. Albeit, that's just a theory at the moment. It may also be manipulating Ami as well:
The 'Rain King' and 'Amy' Happy Events seem to be it's Happy Events, considering the context around them. Probably also the 'Second Sun' Happy Event, considering Long Maya mentions frogs and rain:
Plus Long Maya also seems to be trying to make Sensei and Us question things:
Which seems textbook Pareidolia.
Overall: The connection seems to just revolve around Pareidolia and it's logic.