I'm planning on watching this again for more examples, but I'll throw out my arguments for the throughlines.
I could kind of see "killer on the loose", but this doesn't feel like the Fugitive where an innocent man has been wrongfully accused to clear his name or Shawshank where everyone is stuck in the prison under the evil Warden's rule. Buffalo Bill is a free man, but the conflict is that he is actively stalking, kidnapping, killing, skinning, constructing a suit out of skin, and they can't stop him.
Clarice Starling Universe
She's a cadet, she's inexperienced, a woman in a man's world. She wants to make it in the FBI. Her past haunts her and drives her to want to protect people. The scene in the funeral home where are the men are oogling her and Crawford acts dismissive towards her in front of everyone sticks out.
Hannibal Lecter Mind
He's all about digging into her subconscious, making her remember her father's funeral and her summer with the lambs. He makes her nervous and bullies her (the fava beans speech), and belittles her hopes and dreams with his dismissive attitude. By the end, his attitude about her has changed as he sees the world as a better place with her in it.
Also, he's very mannered and cultured, which makes Clarice feel like the poor, white-trash girl image she's trying escape from. He has very strict opinions about manners. He gets Multiple Miggs to kill himself after Miggs is "rude" to Clarice.
The entire relationship feels like Mentor/Mentee growing becoming an Unlikely Friendship/Respect. Their whole relationship is predicated on putting together the profile, trying to conceive of what this killer is like and they grow closer as he pushes her to put together the pieces. A big moment that pushes the relationship apart is when Clarice lies to him about his transfer, in an effort to manipulate him.