Noticed some dark substance which comes out from faucets. Wonder what it might be and what I should do next.
I would first try bleach rather than vinegar. This looks like it could be mold and bleach kills mold very effectively.
We use the same brand of shower head, ours being hand holdable on a flexible line. We use them for years and don't get that brown deposit. If yours is hand holdable, let it hang down in a bowl of bleach in water. First try 1/8 cup in 1 qt water. Use an old tooth brush to dislodge the deposits on the outside and then rinse it away before you immerse the head in the bleach solution. Or just let it hang in the bleach as is and see what happens.
As the first treatment scrub away the dark deposits with a paper towel soaked in detergent solution, then immerse the head in bleach or vinegar for an hour or two.