When Rose speaks with Dark Alessa in Alessa's hospital room, Dark Alessa makes two different statements about herself and Sharon, both containing metaphorical and literal content. At first glance, her metaphors seem random, put there solely for the reason of "LOLSYMBOLISM". But what if they're not? What if Dark Alessa is tying herself, Alessa, and Sharon together both on a literal and symbolic level all at the same time? The literal connection is obvious, consisting of "She is Alessa/ I'm the dark part of Alessa/ the little girl is what's left of her goodness", establishing Sharon/Alessa/Dark Alessa as three entities all sharing one identity. But it's unclear how "I have many names/ She's not your child, she's hers" connect to each other.
My thoughts are this: Dark Alessa is metaphorically referring to herself, Alessa, and Sharon as an inverted Holy Trinity, an opposite to the Father/Son/Holy Ghost unity seen in the Bible. It's already known from interviews with Gans that the whole movie is wrapped in religious metaphor, particularly the duality of the three incarnations of Alessa, so it's not a huge leap of logic to deduce that Gans wanted Dark Alessa to reference the Holy Trinity in her speech to Rose. Dark Alessa's claim to have "many names" could easily be invoking the concept of the Holy Ghost, which is known by dozens of names throughout Biblical writings. She then immediately makes reference to Sharon being Alessa's "child" (though this is physically impossible), which draws a mother-daughter parallel between Alessa and Sharon. Put together, "I have many names/ She's not your child, she's hers" could easily be seen to mean that the three aspects of Alessa form an inversion of the Holy Trinity, an Unholy Trinity consisting of Mother (Alessa)/Daughter (Sharon)/Unholy Ghost (Dark Alessa). This would work with the literal aspects of Dark Alessa's speech tying herself, Sharon, and Alessa together as three incarnations of one person, meaning her speech is working on two different levels: a symbolic level that ties all three of them together, and a literal level that also ties all three of them together. IMO, this was likely intentionally added to further the film's already overwhelming religious symbolsim.