What to do about holes in tile grout?
Yes, in all likelihood this can be repaired fairly easily. It is a good possibility that these holes are due to poor quality work where the original grout was not fully squeezed down into the joints. You may find other areas that are not yet broken open but also just a partial penetration into the joint.

You will have to work carefully to replace with like style grout material to minimize color differences of the repair. 

If I was repairing this I would dig out around these breakout areas back to where the adjacent grout is sound and well penetrated into the joint. Also clean the edges of the tiles in the opened out areas too. Since there is also the possibility that water and soap scum have entered these holes you would want to take steps to clean away that as well before attempting to re-grout the repair areas.