A single word for emotional connection

Sympathy consists of the prefix sym "alike, equal" and the stem path- "feel, suffer". Common usage differs in most English registers for sympathetic, yet Sympathy is defined exactly as asked for

sympathy 1. an affinity, association, or relationship between persons or things wherein whatever affects one similarly affects the other. [...].

The question is, to a degree, not one of language but philosophy and associated fields, including medicine. The name sympatheic nervous system "can be traced to the concept of sympathy, in the sense of "connection between parts", first used medically by Galen".

Yes, you identify with the person.

identification in psychology

I believe the word you are looking for is empathize, to experience empathy:

the psychological identification with or vicarious experiencing of the feelings, thoughts, or attitudes of another.

I would call it contagiousness or sympathetic reaction. Examples:

I'm very susceptible to the contagiousness of someone else's concert nerves. If I visit a green room, I get nauseous, even if I'm not going to be performing.

When it comes to my sister's panic attacks, I get a strong sympathetic reaction. I'm working on increasing my separation from her emotional level.

Using the idea of the word ‘project’, as aparente001 suggested, gave me the word ‘projection’. I like that word, to name what would otherwise be awkward to name & explain. Projection, in this instance, would make you the ‘receiver’, of what someone else ‘projected’. Thus, primarily taking the ‘focus’ off yourself, making it easier to confide perhaps.