The Columbian authorities are explicitly instructed to capture Liz, so it makes sense they'd ignore her in favour of taking down Booker first. Fitzroy expresses interest in making use of both Booker and Lizzie in a voxophone from quite early in the game (edit: just checked the transcript and she only mentions the false shephard), though I can't remember if she eventually tells the Vox to kill both Booker and Liz in the factory, or just Booker. In any case, it makes sense that Booker is the immediate threat in combat, so even the Vox would target him first, though they could've done with some acknowledgement of Liz's existence.
Why not just have them say something though? It doesn't really make sense for there to be no mention of her. "Don't hurt the girl!" at the start of a firefight, "Give us the girl", "We just want the Girl", during combat? It's simple but goes a long way toward making it feel like a living breathing game world. Elizabeth is a major part of the Fitzroy storyline but she doesn't even get mentioned by her? I don't think Slate mentions her either. Outside of Comstock and a few scripted interactions she seems invisible to the rest of the game world. Why would Lady Comstock ignore her? That whole section is about their relationship and hate for each other why is she attacking Booker but not Elizabeth? That doesn't make sense narratively or mechanically.
It's kind of disappointing that even for a developer that has tried to address and explain ludonarrative dissonance to such an extent as Levine there's such glaringly obvious issues with mostly simple solutions. Elizabeth is such an integral part of the story but in terms of gameplay she's so unimportant that even the game ignores her.