This is precisely the claim of which I am skeptical.
Think of a few bodily functions which we don't find embarrassing: Walking, running, jumping, growing red hair. We've got words for people who do such things (e.g. redheads) and we don't find those words demeaning even though they reduce people to a single function.
None of those things are bodily functions. Running is a bodily activity, not a bodily function. Red hair is an appearance, not a function. Breathing is a bodily function, and I suppose you could argue that being called a breather isn't really insulting (just... odd). But it doesn't really matter.
Menstruater and ejaculater and defecater and demeaning terms. We agree on that, do we not? For the purposes of this thread, we don't actually need to figure out exactly why.
"Mouth breather" is absolutely pejorative.
I think in the case of women, a lot of the objection rises from a history of societies reducing women to second-class citizens, chattel, mere functions. People may or may not object to reductionist identification, depending on a number of circumstances.
Most people don't object too much to being described in terms of a career they chose, especially if it's a prestigious or respectable career. I'm not reducing myself to a mere "firefighter". I'm summing up the core of the self-identity I've built up over many years of effort (or that I aspire to build up, having just been accepted to the department). But even that can be grating, if people apply stereotypes, or forget I'm also a spouse, a parent, a sibling, a volunteer at the soup kitchen. Etc.
But I suspect very few women think of "menstruation" as a summary of their core self-identity. After centuries of being reduced, of being made less-than, women have good reason to say, "I'm more than just a body part or a function. I'm a fully-fledged human being. I have a career. I have a self-identity. I have equal standing with every other human being, whether they menstruate or not."
And the only reason women are now being reduced by society yet again, is because of transwomen. A tiny fraction of the population that suffers from gender dysphoria, saying, you can't say your products are for women, because that doesn't include us. Well, it doesn't include post-menopausal women, either. It doesn't include pregnant women. And yet you don't hear the AARP pitching a fit about period product advertising not being inclusive enough for their membership.