That is called a “No true Scotsman” fallacy.
It is a feminist action because a large swathe of South Korean feminists openly adopted it as a shibboleth, whether it is consistent with the principles of your preferred brand of feminism is irrelevant.
It’d be the same as saying protesting in front of abortion clinics in the US isn’t Christian (except Christianity has explicit rules so the comparison isn’t 1:1) when most if not all protesters would cite their religion as their motivation.
In the Netherlands it’s, no joke, 15 minutes and ~60 bucks.
Did it before, as a foreigner too, easiest shit in the world.
They even have dual language forms.