To be honest that was my first thought as well. Given the fact that Kishi wasted so many panels giving a background on Shin and his connection with the "clones" and the way Sakura replied to him regarding the bond between a parent and a child, that theory is plausible.
People seem to claim that since Sakura have Byakugo she would be able to "fix" her organs and all. But they forget something. Byakugo is accelerating the way the cells are being reproduced but that number is NOT infinite. There is a limit. By using this you literally shorten your life span as Tsunade explained. And some things could be way beyond cure. If you don't have something in the beginning then how are you going to reproduce it?
Now back to the topic. Even if Karin is her biological mother and even if Sakura couldn't conceive or whatever the reason is behind all this, it DOESN'T change the fact that Sakura IS her mother. She is the one that raised her. She is the one being there for her. She is the one that cared for her and loved her. Also she is the one that rushed to defend Sasuke and Sarada from Shin's attack and not Karin. So yeah SHE IS her mother.
Think of it like this. Lets just say that your biological parents abandoned you at the doorstep of some stranger. Then that stranger takes you in, feeds you, raises you and loves you like he / she was your biological parent. What does that makes him / her? This person that took you in, THIS is your true parent and not your biological one. So yeah Sakura didn't lie about her being her mom and acting like one.
P.S. I am actually a Sakura hater. The only female character i liked in the series was always Hinata cause of her shy attitude. But it seems weird watching people saying stuff like that about fictional characters without first using some logic. Its a manga yeah but still, some logic is applied to it.
