You want to be careful from a few viewpoints. First, on a lot of old pieces the “grain” is actually printed on — yup, printed. If you aren’t careful you can remove the printed grain only to discover that what’s underneath isn’t nearly so beautiful.
The second reason to be careful is patina. Over the years wood develops a beautiful patina. That’s why antiques look so much better than modern pieces. Unfortunately patina can be destroyed by most chemical strippers, as well as by vigorous sanding.
OK, so with those caveats out of the way.
If what is coming off is flaky, I’d agree it is likely shellac. Shellac dissolves in alcohol — which is how you can test to see what it is. If the flakes dissolve in alcohol you are dealing with shellac: you can carefully test a hidden piece of the table to see if alcohol removes the finish WITHOUT damaging the patina. If it does, that would be what to use: a stiff brush (of the sort you’d use for oil painting) and pure alcohol, scrub it out thoroughly.
To re-finish, I’d go for shellac again. If you’re dealing with an older piece, best to keep the finish the same as the original. Less likely to encounter compatibility problems.
Those are my $0.02 worth.
You want to be careful from a few viewpoints. First, on a lot of old pieces the “grain” is actually printed on — yup, printed. If you aren’t careful you can remove the printed grain only to discover that what’s underneath isn’t nearly so beautiful.
Ouchy, I did that once and ruined a old piece...:O( a sad lesson learned..