There were many designs of mechanical doorbell actuators : lever , twist/crank , pull , push , chain , cable (& rope) .
This one is a 'trip lever' or 'trip cam' type , with an internal escusion (missing) used for securing the exterior mechanism through a door with threaded machine screws [and sometimes screw/nut combo too] (much like what you'd see in a modern-day exterior deadbolt mounting) .
It was assembled by first mounting the actuator assembly , then the striker assembly , and finally the bell ; all of the mounting hardware would be concealed by the bell itself , making a very clean-looking installation .
I'm not sure if yours was/could be adapted for through-the-wall mounting (I doubt it ) .
Nice c1860-1880 unit , stonewallmaster .