I don't see them fighting for the right to hang laundry. No action has been taken against the woman, and the man was fined by an association (I don't like associations, but they have tons of restrictions that he voluntarily entered into).
If you look at that picture, it seems the woman is hanging her laundry in the front yard. It seems the article is all her side of the story without talking to others. There's a very real possibility the neighbors leaving notes may just want her to dry the laundry in the back yard, which is the normal thing to do.
You have every right to do things that are atypical or even antisocial so long as it's not illegal. That doesn't mean other people can't ask you to stop or ridicule you.
Flagging this because it's a click bait headline with substandard content that seems to be misrepresenting the situation.