Or at least, my assumption was that Tachiyomi is just a gussied up PDF reader, which would make the litigation baseless. If they'd specifically added features for piracy or advertised the ability to use Tachiyomi to read manhwa owned by Kakao Entertainment Corp, then yes, there'd be some basis to this. However, even if it isn't baseless, most Americans are financially precarious enough that merely stepping inside a courtroom would bankrupt them. Having to defend against a copyright lawsuit is out of the question, especially an international[2] one.
As for what's going through Kakao's head? Something between "get off my lawn" and "fuck you, pay me". This isn't a Japanese, Korean, or even East Asian thing. That's just how copyright works, and it's been that way for centuries. If anything you can say it's an American export, because Walt Disney is a horrible person, but that point's already been beaten to death here.
[0] Korean for "manga"[1]
[1] Japanese for "manhwa"[0]
[2] The lawyers in the room will object to my use of the word 'international' here. In reality, either Kakao sues in the jurisdiction of the developers directly, or they sue in South Korean courts and get the judgment domesticated into each developer's jurisdiction. The right to do either is a fundamental part of basically every free-trade agreement ever, so this is an implementation detail.