There's a good probability that the finish is orange shellac. If it is, you can find out easily by test swabbing with alcohol in some inconspicuous place, like the top back side of a leg or underside of the upper table top. If your test removes an amber color from the swabbing, then it is quite likely coated with orange shellac. If the swabbing with alcohol doesn't remove the amber color, then the surface is likely lacquer. Lacquer can be tested by swabbing in a similar way with Lacquer thinner, but try it after testing with alcohol.
If it is shellac, you can either clean it up and maybe apply a fresh coat of orange shellac, or learn how to "French Polish" to renew the original coat. Orange shellac was heavily used to finish furniture in the 40's and 50's. It's quite forgiving and easy to repair, but it is also easily damaged with water. The white rings formed where drinks were placed is an example of this damage.
You can strip most of the shellac off by just soaking and wiping the surface with alcohol. Applying another finish directly over orange shellac is not usually successful because orange shellac contains wax and other finishes will not adhere well. For re-finishing with something else, remove as much of the orange shellac as possible. Then apply a coat of de-waxed shellac to seal the surface. After this is applied and allowed to dry thoroughly, you can apply most any pigmented stain and polyurethane or lacquer over it for a more modern and more water resistant surface.
If it was mine, and my dad had made it, I would do everything possible to restore it to
as original of a condition as I could, and keep it to remember him.
Central North Carolina