One significant detail -- she wouldn't call the daughter she raised a "fake." It would be more like an adoption scenario; she's not the girl's birthmother, but she is her mom. They have an actual parent-child bond, so it is in no way "fake."
She'd want to know her birth-daughter, of course, but at the same time, she'd be concerned for the girl she's raised. She'll want the child she's raised to know she loves her, while still getting to know her birth child. She'd also be in a place of fear herself, on some level, wondering if the girl she's raised will want to leave her, or if she'll prefer her birth family, or if the birth family will even accept her.
(This could be more pronounced if one family shares a belief system or ideals that are incompatible with the other. A girl raised Pentecostal discovers that her birth family are Atheist, or vice versa, for example. The family who has raised her might be reluctant to allow her to see the other family, or likewise, the other family might feel it's their duty to "fix" their birth child by altering her views to match theirs.)
Also consider that family history can have something to do with the reaction. If there's one child with a physical condition, but not the other, or if genetic illness runs in one family and not the other. If one of the mothers was in foster care or adopted herself, it could lead to significant things, such as a fear of being perceived as "abandoning" her child, even though that's not what happened.
If something jarring or traumatic has happened to one child, there could be feelings of "I was never supposed to go through that." Differences in socioeconomics, if one family is wealthy and one isn't would be heightened. The wealthy mother realizes her child has been raised without the privileges she would have had if she'd been with her birth family, while the less affluent mother sees her child as being given an opportunity for a life she could never have provided.
Your story, and the mothers' reactions, is going to depend on the variables in your particular scenario.