The term “embodied empathy” has been around for at least 50 years in the psychology world and it is still a contentious term. Basically, it involves a therapist to immerse his or her self into the feelings of the patient so as to be able to perfectly emulate those feelings.
This is not used solely in the realm of psychologist/patient however. There are people who do this while on a date in order to make the other person feel as if they are understood on a deep level, thereby trusting the other person more than they typically would.
This is why the term is contentious. If used by professionals in a constructive manner, it can be a great tool to help the patient get better. If it is only used in an underhanded way however, this method is being abused.