How to tell what type of finish you have:
Fortunately, you have some flakes.
The finish is water damaged...that's why its lifting.
Drop some of the flakes into a glass jar of denatured alcohol.....if they dissolve, you have a shellac finish.
If not...put a few flakes into some lacquer thinner....if they dissolve, you have nitro or acrylic lacquer.
These finishes are cross-linking finishes; I.e. one layer melds or melts into the next because of the solvent nature of the carriers (the liquifiing agent).
If its neither of these, it could be an oil based finish, or one of the newer acrylic or water based urethanes, which don't lend themselved to easy repairs.
Both of the cross-linking type finishes can often be repaired without full stripping and refinishing....but in the case of this piece which has large amounts of missing finish, I would mask off everything but the top and use a soy stripper like Ready Strip to lift the remainder of the top finish. After you are done, rinse the top with a wet rag to remove any remaining stripper. Let it dry a couple of days.
LIGHTLY hand sand with 320 on a felt covered block to knock down the grain from wetting. De-dust with a vacuum or pressurized air.
To determine what it will look like without stain....take a rag with some Paint Thinner and wipe it across the top.
How it looks when wet will pretty much be how the various woods will look with a couple coats of clear lacquer or shellac.
If how it looks is close to the rest of the piece, there are tints (dyes) available from Woodcrafters that can be mixed in to the finish before you apply to get closer to the shade of the rest of the piece.
If its way off, you may have to experiment with stains to get it to match before you top coat.
P S... I'm just guessing.... But what you have there looks like the darker veneer is mahogany and it appears there was little if no stain..
Most likely you'll end up with just light amber shellac or light amber lacquer as the only company.
But again, I'm looking at it through camera and display and, as they say: you're results may vary......
Yes, Graphus is right....water based strippers do run the risk of lifting old veneers. .....but the fumes of the old stripper can burn flesh and lungs of the uninitiated. Veneer can be repaired. Health; less so.