"Caring about someone" doesn't have the same meaning as "taking care of someone".
It would help if Duo provided context situations for the sentences. Seriously.
why is заботишься in the reflexive when it fits a more accusative description with the "обо"?
Think of it like Spanish "preocuparse", it seems to be very similar. So it has some similarities with English "worry". When you (impersonally) "worry about somebody" you "take care of" that somebody.