Empathy - the ability to understand and share the feelings of another.
Empathy is pretty close, but I get the impression that you're looking for something more literal.
Mirroring - is a psychological term to describe the behaviour in which one person copies another person usually while in social interaction with them. It may include miming gestures, movements, body language, muscle tensions, expressions, tones, eye movements, breathing, tempo, accent, attitude, choice of words or metaphors, and other aspects of communication. It is often observed among couples or close friends.
Again I think that's pretty close, but not quite perfect.
Sympathy - understanding between people; common feeling.
I think Sympathy is the best answer I'll be able to come up with. As argument I present Couvade Syndrome or "sympathetic pregnancy". It is the "proposed condition in which a partner experiences some of the same symptoms and behavior of an expectant mother. These most often include minor weight gain, altered hormone levels, morning nausea, and disturbed sleep patterns. In more extreme cases, symptoms can include labor pains, postpartum depression, and nosebleeds. The labor pain symptom is commonly known as sympathy pain."