from the way she intreduces herself I'm pretty sure it's not her real name. The game itself heavily implies it when it let you have the dialog option "what is this name".
Oh yes, if I recall correctly, Shar's priests go through some weird rite giving them amnesia, but they usually do it when they get old and senile, they give up their memories to Shar. So it could be the reason why she might've entirely forgotten herself, name included. What puzzles me, is that Shadowheart seems to be a too young to have gone through that particular rite. Maybe she knew too much? I dunno.
She says Shar is her mother but It doesn't seem like she means it figuratively. She says "she took me in". it could be figurative but I'm not 100% sure. perhaps she was adopted into a tample or something like that
I always assumed it was figuratively, her being an urchin makes sense that some Shar clergy saw the potential in her and took her in. Also, in FR lore, there was that sect that thought that Selune and Shar are just two sides of the same goddess, two sides of the same coin. I wonder if that will play some part in the story. Especially with the temple of Selune being abandoned... and there are quite a lot of texts pointing that Selune's clergy has gone weird some time before our main characters capture by the mind flayers.
