When someone makes fun of a guy for being "like a girl" in any way, this is a form of misogyny, because it's denigrating the female traits in a guy. Also, when a guy is afraid of being girly, or makes fun of other guys for being girly, this is called toxic masculinity. Toxic masculinity and misogyny towards males is a pervasive part of our culture, and it shows just how pervasive misogyny is, not only towards women, in our entire society.
Some of the other ways this manifests is making men repress their emotions, it's often seen as weak to show emotions if you're a guy. This could lead to repressed emotions and feelings of insecurity and shame, and corrode the psyche of the male mind. It obviously has an effect on the gay community, and you see it in cases where guys say "I'm fine with homosexuality, but there's a certain lifestyle and way of behaving attached to it". They're of course talking about the way that some gay guys subvert gender norms, but they could also be talking about sexual fetishes. Either way, it's bigotry. It could also be one of the reasons why men aren't taken seriously when they say they've been raped, and this is what people mean when they say "rape culture".
There's many other ways that misogyny can effect us all, and this is why men should call themselves feminists. I call myself a feminist because I realize that not only should I try to be supportive of the female species, but I recognize that misogyny isn't just a problem that effects women, it's a problem that effects us all. This is why I say, we need to smash the patriarchy; we need to smash the hierarchical structure of the dominance of toxic masculinity.