However concept art showing Girl's & Robot's interaction implies the opposite. That robot is an intelligent empathetic being that can act of it's own accord.
I'd like to suggest both should be true. Robot would begin as nothing more than a weapon/tool for Girl to use but over time Robot would reach beyond this and become a person in itself, a person Girl so sorely needs in a lonely castle full of evil much more than she does any tool it contains.
This transformation should NOT be expressed through cutscenes/narration (alone at least) but through gameplay.
Robot would start as an inanimate being only capable of being controlled, as seen in existing gameplay footage. However as the game progresses Robot would gain abilities that would make it more and more capable of independence from Girl.
These abilities would progress in a slow manner displaying a gradual change in Robot i.e. Robot can be controlled while carrying Girl -> Robot can lift Girl to high places -> Robot can follow Girl by holding her hand -> Robot can defend itself when attacked without being controlled -> Robot can act defensively when enemies are nearby -> Robot can rush to Girl's aid when she's in danger. All these would express more and more active independence from Girl's control, maybe it could go even further.
Abilities aren't the only way Robot's development of a personality could be expressed. Another means would be through animation. At first would move in a manner mimicing Girl, attempting to use the same animation but unsuited to Robot's body these would be slow and clunky. As Robot grows more independent it would gain a more distinct and eventually unique movement pattern and animations that make Robot's motions much more deliberate and functional in combat. Making weapon draw faster, dodging more precise, etc.
Another sub-suggestion that meshes well with this is to have Girl develop abilities that reflect Robot's development. With Robot's transformation from toy to friend Girl should be less eager to let Robot go it alone in more and more difficult fights.
As such Girl should slowly learn to fight, gaining abilities that can help Robot in fights i.e. throwing rocks to distract enemies -> throwing rocks at enemies to stun them -> throwing water on enemies to slow them down -> throwing bombs to harm enemies directly -> acquiring a slingshot to help with ranged combat -> acquiring a hammer and performing mounted combat riding on Robot's shoulders letting Girl being an extra pair of arms for Robot.
Another way to show Girl's & Robot's divergence is the control scheme, they could start with near-identical basic control schemes that become very different as the game progress showing the development from Master & Puppet to Sisters in Arms.
I don't know how you planned to tell your story, but I'd suggest that The Girl and the Robot would be a perfect game to tell a story through gameplay in.
Edit1: Some more gameplay developments for expressing Robots transition from thing to person;
1). Movement sounds change from repetitive, loud, sharp, & mechanical noises to diverse, soft, & organic noises.
2). Greater caution. i.e. over time Robot will start resist or even refuse dangerous actions, such as swinging a weapon or pushing blocks when Girl is too close. At first Robot will let Girl make her dangerous mistakes, but not forever.