Honestly I could see Jean being the one getting quickly attached in her new-found friendship(?) with Peter.
In BND, Jean is lonely. She wants a family that will accept her the way she is and of course she'll get this in the future with the X-Men... but right now? She is mourning the fact that she almost had that family of her dreams. She almost had a mother but she gave up on Jean so she never was family to begin with. She had a sister who was just like her and loved her a lot. But she was taken away from her. These wounds are fresh.
So, of course it would be Spidey she'd latch on to after the events of the movie. He is also different like she is and he can relate to her struggles! He even shared his maternal figure (Aunt May) with her who gave Jean comfort just like a real mother would have.
So yeah! In my mind Jean is definetly projecting some sibling/mentor role on to him (just a smidgeee)
She is 100% not happy about it tho and will cringe about it later.
