How can I restore the shine to my porcelain sink?

It has probably worn the finish off. Baking soda with a scrub daddy will help cut through any build up. If it's still dull then you will know the finish is depleted. You can try to re-glaze but I'm unsure about that process.

Chances are abrasive cleansers were used along with incessant yet unnecessary scrubbing have eroded the original factory finish. Nothing will bring back the original shine. Don't worry, you are not alone, many people have done this unknowingly. You can have the sink reglazed professionally. Then do not use products like comet or Ajax. Bar Keepers Friend or Bon Ami will work, yet be non abrasive. This also goes for porcelain tubs and showers too.

To put the shine back on you can use Gel Gloss or you can use some auto wax, carnuba, and shine it up with a soft cloth.

you could try a soaking of borax and water overnight or tsp and water overnight

I fill my white porcelain sink with a scoop or pod of oxiclean, hot water and let it sit until the water gets cold. Then rinse, wipe down with a microfiber cloth and a magic eraser for any rezidue. Tough stain? I use Bon Ami or Bartenders Friend.