Taken from /m/

One irony of Evangelion is that Asuka calls Rei a puppet.

Asuka was reared as a small child to become an EVA-pilot, and never ever considered that she’d have some sort of other future than the one planned for her by NERV.
Rei was also reared from a small child to serve NERV, but she ends up questioning not only herself and her own role, but

Practically the consequences of this is best shown in Evangelion’s late-game.

When Rei loses her EVA and cannot pilot any more, she is virtually unaffected and instead turns the focus on antagonizing NERV and understanding her situation better.

When Asuka is unable to pilot because NERV has rejected her, she loses all will to live and becomes catatonic to the point you’d think she was comatose.

We can see here that Asuka is and always was, a puppet being controlled by NERV. Like a puppet, she collapses into a lifeless abhuman once the strings to her puppet masters, the EVA, is cut.
Rei on the other hand, proves Asuka wrong by not only retaining the integrity of her personality, but also strengthening it.

There is a reason for why Asuka is the one with dolls in her background, and Rei is free of it.