BB, I think you don't see why it matters because it's not a direct putdown of your identity. You (and I) never really feel the impact of being in a world where people constantly use "like a woman" (and all its variants) as a generic putdown. Not only that, it's a world where people will go to lengths to defend this status quo, which is precious to them without their even realizing it.
The appropriate analogy would be if people derided other people by insinuating they were like men who had sex with women, with everyone "getting" that this is a shameful suggestion. Pretty ludicrous scenario, I know. It's so far out of reality that it's impossible to actually imagine.
*We do regularly talk about men having sex with women and of course, it gets associated with awe and respect instead -- in direct proportion to the amount of women and the degree to which they're objectified in the course of it. I mean, we come from a tradition of marginalizing women in society. But we have also, genuinely, come a long way from it. We've progressed enough to recognize this s#!t when it happens and know the parts of our tradition we should be letting go of, instead of blindly perpetrating. In particular we are more conscious than ever before of how "that's a small thing", "people are being ridiculous", etc, are and have always been deployed as permission for ignoring/dismissing important, real talk. But sure, this term isn't the biggest deal in the world, and I don't know that anyone ever claimed it was. It is, however, quite emblematic. That shouldn't be hard to see.
We don't get to call problems solved while not living up to the lofty standards we like to proclaim are universal. In similar vein, I'm noticing this rather unfortunate line in your sig: "White privilege is a fallacy." Strength is rising above the easy outs.