I have noticed that a lot of people have misinterpreted Clara’s “I was born to save the Doctor” line and the fact that she was at one point a Gallifreyan woman. Quickly and succinctly, this is who the Impossible Girl is:
Clara Oswald is a pure, 100% human woman born in the 21st century in Lancashire. She has human parents and grew up as a completely ordinary girl. She was not destined to become the Impossible Girl; it was a choice she made in her own right.
What makes her impossible is that she, as the original Clara, jumps into the Doctor’s time stream and splits herself into millions of versions that will be born at the right time, in the right place, to save the Doctor’s life - hence being “born to save the Doctor”. Each echo is similar to the original, with some characteristics amplified and others diminished. She has the same face and a similar name - Clara Oswald, Oswin Oswald, Clara Oswin Oswald, you get the idea. And to link her back to herself or to get him to recognise her, she has her line: “Run, you clever boy, and remember me.”
The echo Claras are drawn to the Doctor, no matter which version of him she encounters, and she will save his life. This doesn’t mean she has all the exact, crystal-clear memories of the original Clara; he’s in her subconscious, a constant presence in her mind that she doesn’t constantly think about but is always there. Perhaps she doesn’t know she’s in love with him, but she will always save him because that’s what each echo was born to do, and the fundamental part of her knows she loves him even if her surface memories don’t.
Victorian!Clara, for instance, was instantly attracted to the Doctor even though she didn’t consciously recognise who he was at first, while Oswin subconsciously loved him so much that not even the Daleks could take that away from her, which enabled her to save his life.
She will be whatever his life and situation needs her to be. If she has to be there to choose the right TARDIS for the first Doctor, she’ll be born as a Gallifreyan because there are no humans on his planet. If she has to be a Victorian barmaid, or a Junior Entertainment Manager on the Starship Alaska, that’s what she’ll be. Just as long as she is there when his life is in danger, so she can save him.
So what does this all mean? She’s still Clara. She’s still human. She isn’t related to him, and she isn’t a Time Lady. She is Clara Oswald, an ordinary and incredibly brave human girl who does something extraordinary to make herself impossible. While not all the loose ends are tied up, this is who she is and why the Doctor kept running into her. She is his “impossible girl”, the one woman who has known him for 1000 years and been there even when he didn’t know it. And that’s pretty incredible.