Piano black polish and refinish

BASF makes the finishes on the Hamburg Steinways. Obviously a vast company and I doubt they sell a "refinishing stick" or the like. There must be a good piano restorer near you-- worth doing a little digging. You in Dallas? Judging by your "handle." Good question. I know people have used high gloss black nail polish or similar but you can see the touch up in strong light. 

To get the quality of finish that comes on your speakers you need the resources of a professional paint shop.  Best for touch ups, IMHO, is the paint touch up kit made by 3M.  The P1200 rubbing compound is quite good.  You normally start with a small disc of ultra fine paper and water, then use the rubbing compound to bring back the gloss finish.  Difficult to not have some swirls vivable in the proper light.  An auto paint shop may sell you some sanding supplies and some "purple" rubbing compound.  

Do a search here for "Meguair's".   Lots have chimed in in regard to their polishes.

I can't think of a more difficult color to match than black.  

The good news is that in the hands of proper cabinet maker, gloss black with lots of coats of lacquer/polyurethane etc, is easier to repair than satin.  BUT it also easier to screw up massively.  I urge experimenting on the back or someplace you cannot see it.  I would hunt down the local cabinet maker who understands piano black.  

Brad