That's where the narrative is going, I think: as time goes on and she is forced to begin constructing "Kaoru" instead of mimicking him, she simultaneously works through having lost him and emerges from grief as a different person in a relationship built on who she's become rather than what her brother might have been. Assuming the plot stretches on long enough Lily too may realise that by the time she discovers what's been going on she's fallen in love with this person rather than the voice from the letters.Tbf, she didn't know what Kaoru likes anyway so it's not like she would have lie to her as neither the other girl know what Kaoru liked too.
the problem is she doesn't know what Lilly knowsShe could've lied why does she feel the need to fully become her brother when Lily doesn't know her brother that well either. There was no reason to blurt that out! I hope this manga updates again soon I need to know how this all plays out!