It's either a very poor varnish job indeed or repeated coats of some proprietary stock oil or possibly even linseed oil. You'd need to look carefully to see what was done before: was the metal work removed before this mystery finish was put on? If so, it might be varnish. Does the finish "splash over" onto the metal anywhere? If so, more likely to be an oil build up put on by hand with the rifle assembled.
I don't see any of the rounding over and loss of detail that usually accompanies Bubba's attempts at stock refinishing, so my guess is that it is just has oil rubbed on. In which case, you might as well leave well enough alone.
Fine steel wool would be the best course of action once the metal work is completely removed, if you're determined to do something, followed by linseed oil. Be very careful about rounding over the edges, as not only will you spoil the profile, but you will "go through" the existing finish and that will produce an inconsistent look when the new oil is applied.
Innumerable vintage arms have been spoiled by over-enthusiastic tinkerers who just can't resist "improving" things. If in doubt, do nothing, though of course the tinkerers rarely have any doubts.
Not trying to be patronizing; please take in the spirit intended. ;-)
Very nice condition rifle by the look of it, deserves careful handling.
