My view: Eccentric Family's second season is about growing up
A lot of things happened in the first season of Eccentric Family, but to the extent that it had an overarching theme and development, I remember it as being about family and the Shimogamos coming to (somewhat) deal with their father's death as they learned about what happened in that crucial time. The result was often spectacular. However, this left plenty of room for exploring both the characters and the setting, and simply having some really beautiful and affecting adventures.
Eccentric Family's second season is different. In particular, I have come around to the idea that it is strongly focused on forcing all of the Shimogamo brothers to grow up and to move beyond the comfortable stasis that they had been in after their father's death (and for Yasaburo, probably even before then). The first season did cause some character development, but looking back I think that it mostly affirmed and revealed character, not changed it (Yajiro being the exception). You can't say the same of the second season; by the time it ends, Yajiro, Yaichiro, and especially Yasaburo have made major changes to the paths of their lives.
The Shimogamo brothers are not the only people to grow up and change, either. Almost the entirety of Benten's arc is about cracking her shell of omniscient invulnerability, and the last episode reveals that the Nidaime is also stuck on the past, unable to move on. All of the major characters in the second season need to grow, and all of them get hard shoves about it. No one gets out unchanged and unaffected.
(The one prominent character who refuses to grow and change pays a heavy price for their inability to let go. Twice.)
Things with Yasaburo and Benten especially stand out to me. In a way it would have been easy for the show to let Yasaburo continue through his life with the crowd-pleasing ambiguity and indecision that he showed in the first season and much of the second. The interplay between Benten and Yasaburo is always enjoyable and great, and they have such a complicated and deep relationship that things could have continued for a long time. But the second season does not let the situation stand and in the end Yasaburo is forced to get off the fence; it's clear that his relationship with Benten will be quite different from now on.
(I mentioned this on Twitter and I feel like saying a bit more about it, even if this is not as coherent as I had in mind when I started writing.)