As a once-angry young man who mellowed out somewhat (I am now an angry 30-year-old man), I do understand some of the prickliness involved, even if it doesn't apply to me anymore. I was always pretty liberal and anti-manosphere, but there is an element here that isn't "Men always have to butt in on subjects where we should be listening to women" (that definitely IS a problem, mind).
We, as men, are socialized to deal with othering in the most dogshit ways, and like rubbing salt in a wound, inevitably aggravate it. You don't talk about getting othered, unless you're getting angry about it, otherwise you're 'weak' and need to 'nut up' and 'stop being a pussy'. You can't work to solve it, because then you're a 'tryhard' and 'pathetic'. It's a kind of helplessness by being stripped of the natural tools that should be available to us, but generations of toxic masculinity have rendered anathema.
It's like being trapped in a cage, where you can see every piece of what is tormenting you, but do nothing about it except grind your teeth into dust trying fruitlessly to chew through the bars until some power, through no influence of your own, releases you. No one wants to be othered, no one wants to be seen as fundamentally contrary to participation in a common community - but many men have no way of dealing with that, and it terrifies them. The wounds never heal, but you become increasingly defensive and neurotic about it. It becomes a hair-trigger.
A lot of young men right now are probably reading the bear metaphor as more an incident of othering rather than an expression of the risk inherent to women when dealing with our current society. They aren't hearing "Jesus Christ, be a little receptive to the concerns of women, the risk calculus here is not the same risk calculus you are using", they're hearing "Women don't see us as equals, they see us as dangerous animals. We're not of a common community; we've been (or are being, or are realizing we've always been) cast out."
Obviously this gets the dander up on misogynists, but even many otherwise-feminist-leaning men will feel hurt by seeing it this way. And the reactions of some individuals - using that same 'nut up, pussy' toxic masculinity dialogue, but in 'defense' of a feminist metaphor - is twisting the knife, putting those who understand toxic masculinity back into the intensely frustrating position of trying to explain why that's a dogshit response, and making those who don't understand toxic masculinity double down in the natural, automatic reaction that they've been conditioned to embrace in response to being othered - pain. And from pain, anger.
tl;dr; The reactions of many men to the metaphor are problematic, but it's not as simple as "Bunch of sexists are unhappy that they have to consider other people" for all of them. A lot of is "Bunch of broken men are being given the exact scenario they are used to exercising their society-approved maladaptive coping skills in, with both sides effectively cheering their response on as it serves their own prejudices and preconceptions."