If the mold/mildew got into wood, it will keep coming back, as the mold will grow 'fingers' through the wood pores and you won't be able to kill it. Yes, you can wipe down the surface but it will grow back. The black could be mildew or it could be black mold.
I wouldn't chance that it isn't black mold and would dispose of it. Our storage didn't have a leak, but it wasn't climate controlled and it was in a humid area. We were waiting for our place to be ready, so stored our furniture and household items. I feel for you. The bottom of my stomach dropped out when we first saw it, and everyday after while we were going through everything. We ended up losing a lot of personal family items, and replacing pretty much everything else. It was hard, both financially and emotionally.