I like your thinking and I'm all for Eirin being adopted by Richard's family since if she went back to the Etam family she would be mocked for being a lizard shape-shifter by the rest of relatives and her dad wants to make her the heir meaning more hate and objections from the family. even though her father and brothers care for her, the rest of the Etam house doesn't. so it's better to go to the Kollin family since the young master is obsessed with her and loves her a bit too much XD....and since the Richard's father is a fool for his son and they don't have prejudice against lizards. then Eirin will live a comfortable lifestyle.
but the perfect scenario (which is what I want in my imagination) is her being an Etam (so the dad won't go crazy) and marrying Richard when she's older. so she will technically be in both families.
Okay so while I agree it wasn't ok to adopt Ayrin without her permission. I do think the circumstances are important to keep in mind. Since Ayrin is a shapeshifter she has zero rights unless she has an owner or family that she belongs to. That means for Ayrin to be able to do anything (like get the house she wanted) she would need some sort of connection/protection. With that in mind, I think it would also be highly unlikely for Ayrin to get adopted (and if she did probably by people with bad intentions). I think the duke was originally thinking of adopting her as a means to help her out (he says he wants to make her happy, and this was when she implied that her family kicked her out and she didn't have anyone to rely on). Also, it helps keep his kid safe because Ayrin would be used as a way to target Richard (with adoption he can help prevent that from happening and keep them both relatively safe). With the adoption, Ayrin gets more opportunities and can choose whether or not she wants to stay in the family. Also depending on when he submitted the application, the kidnapping plays a huge part in it. If he hadn't adopted Ayrin and she was sold to someone, there would be no way to get her back even if they found her. It also would allow him the ability to search for her more thoroughly after her disappearance (he's not going to be given the same leeway and opportunities searching for a slave vs searching for his kid). If he was just doing it for Richard, he wouldn't have adopted her and just forced her to stay as a pet or maid. He also did this not knowing she had a family who cared for her (but at the same time, one would assume if the family actually cared they wouldn't have driven her out).
And if we are being for real, I still haven't forgiven Ayrin's dad yet and would like to see him work for it. He only started kind of caring the last couple of chapters and I think a lot of people forget that he was planning on throwing her away when he got bored in the beginning.
I think a dad off, in which people spoil Ayrin is just what that poor girl needs.