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".