Freaky Nikki is still just Nikki.
She behaves like someone possessed.
She talks about who she is when she's desperate for Bear's love as that being a different person.
And when she's on the phone screaming, it's pretty clearly supposed to imply this is what it sounds like inside of her head.
But, the most important line of the whole movie is: "Just because you chose this for her doesn't make it less real".
He doesn't say "what difference does it make" or something to that affect. He makes it clear that her undying desperare inhumanly extreme love for him is very much real, she's not just some robot puppet.
She's just still who she was but suddenly overloaded with an impossible level of love for somebody. When she breaks down saying she can't just be Nikki, that's not some demon literally explaining she can't be Nikki. That's Nikki explaining that she has to suppress her wants and thoughts and feelings because they're in the way of her loving him more than anything in the world.
She's like a drug addict that's addicted to Bear despite deep down wanting to quit, or someone deep in a cult/religion that they deep down know is bullshit but can't accept what letting that be true would do to her. She can't handle the insanity of both wanting to be who she was before he wished for this and who she has to be for him to feel the same way about her, so she basically has a psychotic breakdown where she looks at herself as two different people. But she's not ACTUALLY. She's just an extreme version of someone who's completely lost touch with who they were before becoming all consumed by needing to be whatever she needs to be to earn someones love, who knows that and handles it by talking about herself like the part of her that knows the truth is some other person in her head, flipping back and forth between treating those thoughts like a moment of clarity vs suppressing those thoughts.
The same way a drug addict trying to quit can have all these thoughts about why they want to and how they're not themselves anymore but then just not feel strong enough to power through the insanity of withdrawals