Well, Polish does, this course just failed to teach it. Since recently, we require the only really correct answer that is "wears". More info here: https://www.duolingo.com/comment/27628688
I would never say "Noszę czarną koszulkę" to say that I am wearing a black shirt at the moment.
I don't know about other Slavic languages, and I think this may have not been taught in the Duolingo Russian course, but I remember from Russian Rosetta Stone course a construction "На ней юбка", which seems pretty similar to Polish "Ona ma na sobie spódnicę" (for others: that's "She is wearing a skirt").