I think you are overthinking it. Remove all context and think of a child asking this to a close relative:
"Do you like my dad?"
"Don't worry, I like you much more."
In other words, Azuki thought that Lily was opening to her (because of course she did, that's what she wants the most), and worried about not being liked, so she tried to reassure her saying that she loved Lily more than anyone, and she wouldn't need to worry about her being liked.