Not sure that's our problem, either. Certain stereotypically feminine things may be stigmatized in men, but I think that is part of a larger problem.
We (society at large) don't allow men to experience, or at least express, the full range of human experience or the emotional reaction to it. We don't allow men to have fun, do things for themselves, or enjoy many aspects of life without guilt. We don't allow men to grieve, or show their suffering. We don't allow men to show a full range of emotion when appropriate (while allowing women to manipulate others using those emotions when inappropriate). We don't offer men empathy when they need it most. We use and abuse men, and give them little to nothing in return.
This is why I've said before, and will say it again, a man doing women's things, whether it's wearing the clothing only they (women) are allowed to wear, or engaging in activities only they are allowed to partake in, is not rejected for degrading himself, but for claiming something that isn't his. He is grabbing privileges this society does not think he deserves. He is overstepping his social status.
The stigma against men doing these things is how society keeps men in line. It is the method used to get men to comply. These stereotypical restrictions, as well as privileges, used to go both ways. Men had their place, and women had theirs. Each place had it's pros and cons.
Feminism (the movement, distinct from women) meticulously dismantled the restrictions on women, and gave women access to all of men's privileges. At the same time, loosening as few restrictions on men as possible, and giving men access to as few of women's privileges as possible. (Whether this was malicious or an oversight is another topic, what matters is that it happened.)
Men are hardwired to provide for and protect women (and children), and this helped allow feminists achieve their goals. Feminism also focused society's attention on only men's privileges and women's restrictions as problematic or even existing. That effectively gaslit society as a whole, and the feminist view of society (as being run by men, for men, at the expense of women) became the dominant one that the vast majority of people in Western society today take as gospel truth.
This is why skirts for men is a men's rights issue. It's a visible example of the absurd double standards we are living under. Is it the most serious, the most problematic? Of course not. But it is one that exemplifies the wider issue. One that everyone can see.