It's copying / parroting.
In the smaller picture, everything a person does, can be deconstructed, into small details, like abstract ideas -- hair color, intonation, rhythmic gate.
You do not strictly have to be empathic, to appear relatable, although just-pretending is being a poser, conformist, or fraud.
I believe that autistic people can get caught-up, in the emotional status of children, animals, and plants, but they focus-in, on one thing, only. Were you watching the road, or balancing your checkbook, answering the doorbell, or burning the dinner.
But, anyone with a laser-like focus can also copy a person, in the same way you might remember trivia or make sketches from memory.