It's certainly an induction coil and may be an electric 'shocking coil' - once believed to be therapeutic. It looks rather small to be a sparking induction coil, but it's difficult to be sure from the photos. Comprehensive info at
https://en.wikipedia.org/wiki/Induction_coil .
I'd be undoing those brass wood screws and whipping off that wooden plate from the base to see what connections lie within.
The coil itself will have a primary winding - probably fairly robust wire, and a secondary of many many turns of fine wire. You'll be very lucky if the secondary is still complete.
The primary circuit is completed through the contact breaker just like a traditional electric bell trembler and is probably designed for a battery from 3V to 6V. I'd first work out which connections go the primary and try wiring it up via the contact breaker (needs cleaning with fine sandpaper). Once you've got it buzzing, see what comes out of the secondary.
Martin