I am writing a fantasy story and my MC rescues a preteen girl who then stays with her for the rest of the story (which is several years). At some point, I feel like it would be in the nature of the girl to ask my MC if she can call her mom, but I am unsure how to go about it. I want to get the feelings of the teenage "adoptee" right, even though it will be told from the perspective of my MC and I don't have a lot of reference.
She may or may not go the route of calling her Mom. That is a very individual experience. Did the girl have a mother she knew previous to this arrangement or was she an orphan? There is no hard and fast rule for this. It will depend completely upon your girl character and her personality and how she interacts with the would-be mom. Disclaimer: I am adopted but it took place at birth. That is a different dynamic than the one you describe.
She is an orphan by the time she is with the MC, but she did have a mother within the period that she has memory. The teenage girl tends to grab for connections with people very strongly in general, not exclusively with the MC, though she is closest to her.
I think it might work best if the girl accidentally calls the MC "mom", then asks if it's alright to call her that.