It's definitely veneer over particleboard or MDF of some kind. That was the "french country" or "fruitwood" furniture that was widely sold. Fake flyspecks on the finish, a generic tan pecan color.

Here's what I would do ...

Clean it: use mild soap and water and a rinse, and then mineral spirits to get rid of all the hand crud and old dirt. Let it dry thoroughly.

Scuff it up with fine steel wool or those green scrubby pads.

Use General Finishes Gel stain in a matching or slightly darker color. Use a cloth dampened with mineral spirits and wipe the stain on in THIN layers, ending with a smoothing pass with the grain. It could be cool to use two colors on the top to make the triangles more obvious.

Patch the chipped spot with a wood filler that is close in color to the final stain, then use a bit of stain as if it were paint to make some fake graining marks and blend it in.

Finish off with a few coats of General finishes wipe-on gel topcoat.

You can buy veneer close to what's on the table. Use a box cutter or exato-knife to cut patches larger than missing pieces. Hold patch over damaged area and carfully trace around patch. Cut marked area out,apply yellow glue on mdf,insert patch and carfully wipe off glue squeeze out with damp cloth.