Her identity is fake in the sense that there was no record that proved that Rei ever existed on earth, she had false papers and no recorded family link, besides being an Alien from another dimension. So she didn't take the place of another Rei the same way Yuno killed Her double in this dimension to take her place.
Ah, I see.
I still see no reason to doubt that this is the 'true' Rei. Maybe she's able to travel between worlds freely from now on, to meet up with the friends she made during her time on Earth. I hope a second season (if we'll ever get one) will tell us more.
Well, it could be that the Rei in the stinger is this world's Rei, only with a different name. After all, the authorities couldn't get any record from the other world's Rei because they were using false data (or something like that).
I think a second season is possible. If the crow is to be believed, there are other Alones somewhere. And if any one of them is as evil as the crow...
That would signify that there are parallel universes like in
Mirai Nikki, but I doubt it because then at least they could have found this universe Rei by running her picture in the Database. You know like when they found Yuno dead body in Mirrai Nikky while Yuno is alive.
So either the "Them" made another Rei for this universe, which is sad because it ain't "Rei" anymore, or Rei have chosen to come to this Universe.
That crow was freaking evil, I am scarred of those birds now, dude was a major douche. Considering how efficient the "Them" were are keeping the Crow from going solo you bet there is a plausible other Alone out there for season 2. Although the same Evangelion type "New enemy each week" might become a bit boring.