I'm very allergic to mold, among other things (all airborne, no food allergies). I also adore sourdough bread. If I were to make my own sourdough starter so I could bake sourdough regularly, would it a) be similar enough to mold to set off my allergies, or b) be likelier to attract mold than anything else in my kitchen?
Sourdough starter will grow mold if it starts going "off". If it's healthy, it will naturally prevent mold from growing, but if you forget to feed it for too long and/or the container it is in is dirty, it can start growing fuzzy stuff.
Feed your culture regularly, and transfer it into a clean container now and then, and it will be fine.