With Amagi's suspicious look+response at the mention of the artifact machine heart, I wonder if she actually already has one and just hasn't told Liam, basically because she knows if he had a reason to think if she was "really alive", he'd be even less willing to distance himself from her. She knows that it's in his political interests to not be so heavily entwined with an AI for his political status, and is trying to keep him at arms length.
(edit: And also it would explain why she just straight-up denied that it was possible, despite what the butler said, rather than just being like "Finding a real one would be hard / expensive" or something similar, because she's trying to nip any ideas in the bud)
It's not explicit, but checking back it looks like there
is a bit where Liam mentions choosing even her "inner parts" (and showing a shot of her chest cavity), which would have given him the chance to accidentally get her one as a kid and not realizing what he was doing, beyond just getting her high-spec parts that she wouldn't otherwise have if she was just a stock maid. It wouldn't be surprising at all to figure out he actually got her one but just forgot, because as a kid he just thought "oh cool heart, neat" and forgot all about it because he never sees it.