Making sourdough starter with bread flour
The difference between flour marketed as Bread Flour and flour marketed as All Purpose (at least in the US) is the type of wheat from which it is milled, and therefore the protein percentage in the flour, a p/l value close to 1 which gives proper plasticity to the dough, and a large enough falling number to allow long fermentations.  

At least the US, all flour marketed as either All Purpose or Bread Flour should be suitable for bread making, although bread flour will permit greater gluten development due to the higher protein level.

The yeast or bacteria in a bread starter eat the starch (after it has been converted to sugar by enzymes present in the flour), which is the bulk of the grain, whether the wheat is winter or summer, hard or soft, red or white.   

Also, the way the bag is labelled is as much a marketing issue as a technical issue.  For example, in the US, King Arthur brand all purpose flour is nearly as "hard" (high in protein) as other common brand's (such as Gold Medal) bread flour.

Use either all purpose or bread flour, as you choose.