Hey there, bluespreadsheet!
So, part of the problem is that it sounds like you are thinking about the penis and the vagina as anatomical equivalents to each other. But they aren't.
Rather, it's the penis/scrotum and the clitoris that are anatomical equivalents: in fact, both of those organs start out the same structure in utero, and then only a little later on in a pregnancy, do they usually either change into one or the other. And if and when you can visualize the whole of the clitoris, not just the two external portions, the glans and the hood, it's pretty easy to see that the whole clitoral structure and the penis are actually very similar. If you want some help visualizing that, you might take a look at the diagrams in this piece --
https://www.scarleteen.com/read/bodies/ ... terus-more -- or this one --
https://www.scarleteen.com/read/bodies/ ... every-body!
The vagina, on the other hand, is something there is not an equivalent of. As well, "penis" is only the word for the penis, just like clitoris is only the word for the clitoris. The typical words people use for the penis don't refer to the whole of that genital system either, just for that one part.
The phrase we have in sexual healthcare for both vulva and vagina are "the vulvovaginal system." We don't even have a phrase like that for the penis, scrotum and testicular equivalent, we only have "the reproductive system" -- some people say "male" but that's not something we do here because we don't arbitrarily assign people or their bodies gender -- a phrase we use when we're talking about the whole of people's genitals and internal reproductive organs.
In terms of other slang for the vulva and vagina, I agree that a lot of those words are used as insults or slurs, but you know, I think that's true of a lot of words for all kinds of genitals -- like, say, dick or asshole. Heck, pretty much all of us use "fuck" as an insult or curse all the time. There's just really no escaping that. So, sure, people have also called people things like cunts (though I always like to re-up, "I'd call you a cunt but you lack the depth and the warmth"
), twats, fannys and the like, but I wouldn't let that stop you from picking any of those existing words should any of them speak to you. People used "queer" as a slur for hundreds of years, and that's a big part of *why* queer people reclaimed that word for ourselves.
Personally, I'm a "cunt" and "cock" person in my personal life, rather than at work where I use more clinical terms, so if you wanted someone's casual 2 cents on a word that works for them, that's mine.