Why is their identity more important than ours, I wonder?
I know you have a good working answer to this question, but I can maybe expand on it a bit by tackling it from a different angle.
It's because words have power.
If you can make someone use your system of words instead of their system of words, that's an attempt at dominance. It's an attempt to strip out the mindset you've already got, and replace it with theirs.
And so much of our minds is basically, when you strip it right back, all constructed out of words;
We all have internal experiences. We all have to find a word for those experiences. We all choose the best fit out of the words available to us. The words we're using in our own heads to describe our experiences are a fundamental part of what makes us unique individuals instead of the Borg.
And we often latch on to a word, as a descriptor for some facet of a personal experience, that's ultimately wrong enough to cause cognitive dissonance and trip us up.
I've had a lot of problems on that front thanks to early decoding skills and using books as an escape. Most of my word-understandings operate on a pure-word level, with the relations between words determined purely with reference to where I read them. I haven't actually linked them properly, internally, to something I have direct experience of.
As part of getting my head round my traumas and not being repeatedly sent into a downwards spiral of despair, I've had to go quite deep into proper feminist analysis. I've picked up a lot of words, I've interpreted a lot of my experience through it, it lets me co-exist with my traumas instead of being flattened by them.
These words mean my trauma is no longer dominating me.
If I succumb to outside pressure, reject the lens of this particular system of words, and replace it with a directly opposing system of words, I can't interpret my trauma in a healthy way.
So anyone who's trying to colonise my mind-built-of-words by forcing me to adopt a different set of words - they're trying to dominate my mind by reformatting it in their image.
If I submit to this colonisation, I have to interpret my experiences through their lens. That leads me to understand myself as a person whose internal experience is characterised as "transman".
Forcing me to interpret myself as something that needs to mutilate itself and force everyone in the vicinity into a subservient position re: opposing mindsets. Fucking me over while turning me into an abuser.
Which, by a roundabout route, means I can drop in this quote I spotted on an old thread a few minutes ago, that jumped out of me. It's a high-up from Stonewall being quoted, probably Ruth Hunt, and the thing that jumped out at me is: I actually believe her statements are true.
She said what trans people face is misperception.
She said that this is like the fight for gay rights in the 80s.
What is a misperception? A mismatch between one person's word-mindset and another's, that means they are unable to communicate properly, because they are using words to mean different things.
And it is like the fight for gay rights, very much like it - except in the 80s the two warring systems of word-assignation were pro-gay-feminist and anti-gay-sexist. The pro-gay lens was by definition an anti-sexist lens. The people against it were therefore trying to maintain patriarchy, sexism et al. The anti-gay side was the sexist, pro-patriarchal-status-quo side.
Whereas the opposite is true with trans-words. One position reinforces the patriarchal status quo, and the other challenges it. And the reason we've got this massive argument over which side is which is all rooted in the human tendency to struggle to think outside the parameters of our word-systems and the fact that most people haven't actually sat down and worked out precisely what it is that everyone else is pointing at when they use the label "patriarchy", or the label "sexist" for that matter.
So: their identity is more important than ours, because their identity is constructed out of the dominant, sexist, patriarchal wordsets the mainstream is constructed of, and their assertion of identity reinforces the propagation of those wordsets into people's minds.
Tl;dr: their identity is more important because their identity is consistent with and reinforcing the dominant patriarchal mindset. And it's really hard to step back from the neverending human obsession with dickering over the precise way a word should be used and properly focus all efforts on the problem (in this case, the dominant mindset in patriarchy). Because we're all, on a fundamental level, trying to push our word-sets out there so we can communicate our internal experiences.
The answer there is to analyse and deconstruct the wordsets to find the common ground in people's heads, and find mutually agreeable words that work for everyone. But that's not possible if one side is an oppressor with the oppressing tendency to dominate instead of collaborate.
I could ramble on for a lot longer on this topic, but I'm wary of a massive derail on a support thread. And also I have to go to therapy in half an hour and I need to put some trousers on. And I've just realised one of the things that stops me talking openly about my trauma is the worry that the people I'm talking to are going to try to get me to interpret it through the lens constructed out of the language of the oppressor and I'm scared of leaving myself vulnerable to an attempt at mind-colonisation by the patriarchy.