What lore do you ignore?

[Cilia]

She also screams "his body and soul" belong to her. Not a super normal thought to have towards a parental figure.

Also fiction always has a reason to include everything. There's no reason to interject a scene where someone orders a subordinate to cross a professional line, such as meeting him alone in his quarters, if there's no subtext. And if it meant something else, they would have clarified that too, in that or a later scene. Instead later she says that.

The appropriate "subtext" requires a little bit of context; in a military setting, you never reprimand an officer in front of the soldiers they command because doing so could undermine their authority among the troops. With that context in mind, I always read the scene as Gaius going to chew Livia out for the Waking Sands massacre - the only officers higher in rank to Livia are Nero and Gaius himself, so he can't exactly discipline her in the open (and based on her dialogue with Nero after the Ifrit fight, she doesn't much respect him anyway).

To Gaius, Livia was just another orphan whose potential he could exploit in the pursuit of his conquest; Livia is possessive of Gaius because he's the only thing that gives her life meaning. There's definitely some unhealthy romantic (and plausibly sexual) obsession there, but it's entirely one-sided on Livia's part (which was confirmed by the Sorrow of Werlyt storyline).