Shellac would react with alcohol, so you can gently try some Denatured Alcohol in a hidden place on the chair to see if the finish comes off easily.
Lacquer is not that old of a finish but the chair doesn't look that old in the dowel contruction methods that were used to put it together.Same trick, but try Lacquer thinner to see what happens instead of Alcohol.
If neither of these solvents really attack the finish, then it's most likely a Varnish Finish.
Agree on using existing parts with new dowels or Tenons to repair the chair, then you won't hardly need to worry about the finish matching a bunch of new wood.