Well, honestly, only Livia actually cares about Lily. The other two only like Livia, and want to spend time with Livia regardless of who else is or isn't around. I wouldn't say they're all friends. Livia is the common denominator.
What would be ideal, is if Lucian gave af about Lily since he's the country's crown prince. A little princess from a foreign country can't do much for the Count's family good or bad. But they don't know Lucian favors Livia yet. Only thing that would make the Count jump ship would be to see the link from Lily > Livia > Lucian > Emperor. THEN he would be all for Lily befriending the princess in hopes that she can also get close to Lucian.
But right now, the Duke has more connections that'll make or break the Count's house. One word from the Duke, and it's game over for the Count. That fact is literally glaring at him. And all they know is Livia is just a "silly foreign princess who is only in the kingdom bc it's a tradition" and Lucian is "an antisocial, cold prince who shows no interest anyone or anything". And then Milan is "abused illegitimate heir who no one gives a shit about... except silly women". On paper, none of that is helpful.
He's a sorry excuse of father, don't get me wrong. But I'm sure he's constantly wracking his brain on how to keep this new nobility, and legacy that he's trying to build. Choosing the strange foreign princess over the local Duke is wild.
Lily's dad is an idiot! Wouldn't it be better to be friends with ROYALTY of not only their kingdom, but two additional kingdoms!? I mean, I'd think having true friendships with ROYALTY of THREE different kingdoms would be wa~y better than just a fake "friendly" relationship with just ONE duke. ┑(￣Д ￣)┍
He should be proud of her.