My new coinage: Echothymia.
Echolalia is a psychiatric symptom, involuntarily repeating another's speech. Echopraxia is the involuntary imitation of another's movements or actions. But the woman we were discussing today imitates the mood or affect of the person speaking to her. This is not uncommon, as we all do it a bit in conversation, and demented folks do it more. Children do it sometimes, though not as often as we'd wish.
But there wasn't a word for it until now. We got by with locutions about mirroring and empathy.