I agree with you. I mean, I’m sure Elsa would still would want to be close to Anna even though her powers would get in the way and they keep her from her sister.
I can imagine that setting on Christmas, like maybe Elsa would leave a present for Anna under the tree. Maybe that’s how Anna got the Anna doll.
You see when the girls share the room they keep their stuff on their side of the room and when Elsa leaves she takes all that stuff on her side. Elsa has an Anna doll while Anna has an Elsa doll.
But later during Do You Want to Build a Snowman, Anna is playing with both dolls. So Elsa could have given it to Anna.