I remember when I was a young trans person and it was still acceptable to refer to oneself as “female-bodied” or “male-bodied”. Unfortunately, way too many people were saying things like “I don’t want male-bodied people in my bathroom!”, and we lost this valuable and useful term.
Then, everyone collectively switched to the much more awkward “people with” language. Unfortunately, people took this as an opportunity to start saying things like “people with penises can’t experience multiple orgasms!” (actually, many trans women and some lucky cis men can) and “people with penises can’t understand how degrading some of these sexualized depictions of people with vaginas are!” (because trans women are known for never experiencing degrading, sexualized depictions in the media).
I have heard murmurs declaring the “people with” language to be problematic. While on one hand I’m happy (dear God that language was awkward), on the other hand I fear that we may soon lose any language to talk about biological sex at all.
“Assigned male at birth” is another repeat offender. One’s birth assignment refers to the social process by which one is declared a male or female at birth. While certain sex characteristics are limited to people of certain assigned sexes at birth (while a few people assigned male at birth have uteruses, none of them function), people of a particular birth assignment can have a wide variety of different sex characteristics. A person assigned female at birth may have a penis or a vagina; they may have a uterus, may have had it removed, or may never have had one at all; they may have a testosterone-dominant or estrogen-dominant system; they may be fertile or infertile.
You may say that these people are outliers. However, there is already a word if you wish to talk about assigned female at birth people without including outliers. That word is “woman”.
“Assigned male at birth” is useful terminology for discussing certain social experiences, such as experiences of sexism in childhood and the pathologization of gender nonconformity among those assigned male at birth. It is not useful for discussing sex. It is also not useful for discussing many other social experiences, such as street harassment, for which the terms you want are “men” and “women.”
Some people understand that you are not supposed to refer to trans men as women. However, instead of understanding the actual differences, they proceed to ctrl-F their vocabulary and replace “woman” with “person with vagina” or “assigned female at birth”. Once enough people do this, then trans people (quite naturally) start objecting to it, and then we can’t say sentences as natural as “I loathe my female body” without someone stopping by to point out that our body is male because we identify as male.
I really, really wish we could stop playing Misgendering Whack-a-Mole. Unfortunately, that’s not going to stop until certain people stop misgendering trans people. Seriously, guys, cut it out. Think about what the words you’re saying actually mean instead of doing cargo cult political correctness, and maybe someday we’ll have useful words for biological sex again.