It always depends on what type of material the fixture was made out of. But your problem reminds me of fiberglass tub and marble molded sink. What probably happened, some plumber used puddy as a sealant on the drain openings. Puddy and the fixture are both made out of petroleum products. Even though the fiberglass tub is made of glass, the surface is coated with acrylic. The acrylic finish is petroleum based. The oils in the puddy break down the coating and cause it to develop cracks. The cracks would normally be invisible, but they fill up with soap scum which blackens after a while. These cracks are not repairable by the normal homeowner, but can be fixed by a fiberglass repairman. Check with your local plumbing wholesaler for the name of a repair man.
You will have to remove both drains and replace them, but it can be repaired. Make sure to use silicone instead of puddy after the repair.