Does the body have any electromagnetic field in the fingers capable of making a current via induction?

Answer 1

Fingers are capable of changing, or disturbing, changing electromagnetic fields. That's how fingers are visible and opaque. That's how fingers are detected on capacitive keyboards and elevator buttons. The body doesn't generate any magnetic fields, and only miniscule electric fields.

Answer 2

Yes, There was a documentary about electronica pioneer Bruce Haack where
he played music by waving his hands over
a subject connected with electrodes on
"What's My Line"