The engine stamped pad is as such.
Tonowanda March 19th 396/425 Special high performance
Does the 162171 tell anything that might lead to indicating what style body?
There is no known additional information other than what you have already.
The mechanical solid lifter engine was only installed in race cars, so odds are it wasn't put in a wagon. Usually it was put in a stripped down, radio and heater delete two door Biscayne sedan body (because the sedan was lighter and stronger than the hard top body and the Biscayne was lightest body sold due to lack of sound deadening and other insulation). Most where painted white when ordered with that motor and transmission.