I am purely speculating on this, but. These flea market people know how to make furniture look good and show good. Sometimes this is only for a short time.
I am guessing that they used an oily stain over something that has been sealed and the stain cannot and will not absorb into the wood because of the sealant. And this might be a cosmetic scratch hider to make the furniture show better and for you to buy it faster.
No problem though. I would remove the gunky stuff from the furniture and then get a Formby's Furniture Refinisher Kit found at home centers and procede to repair and restore the wood. This is not a strip down and re stain procedure, its merely to restore the finish you currently have.