part of identity is our relationships to other people.
I wouldn't agree, simply because I consider relationships as existing between people, not within them individually, and more as 'facts of the matter', as opposed to immutable aspects of individuals themselves. But again, this is simply a disagreement on the definition of "identity". I'm not saying your definition is wrong, but it obviously is different.
A familial connection is a fact about someone's lineage, but it is no more a part of someone's identity than to the extent that that individual chooses to make it so. If I was adopted and have never met the woman who birthed me, then yes, she's still my mother even though I never knew her. But that being a fact has no inherent relationship to my identity. The same is true if I was raised by my birth mother but am now estranged, and she has no part of/in my life—she'll always literally be my mother, but in this case, her existence is no part of my identity any longer.
Nonconsensual trauma that alters one's sense of self against one's will is the only thing that muddies this water at all, I think, but even in a case like that, it is only from within that whatever degree (whether zero or nonzero) those events shape one's identity, can change.
Okay, so if I want a single word, "transgenderality"? That really just sounds bizarre, I have to say. Not to mention I've never seen any person talking about trans issues ever say/write that.
That's only one way that suffix is used, and it's assumption on your part that when you see that suffix, that that's the way it's being used. In other words, I think you should allow for the possibility that it ain't that deep. Was it not obvious from the context of what/how I was writing that I wasn't coming from a transphobic place?
-ism is used for all sorts of nouns that simply describe a state of being (e.g. autism, alcoholism, absenteeism), and that's all I aimed for. And from what you said in your comment, it seems like this is uncharted territory, if there's no actual single word term regularly employed for this particular state of being—all of your examples are multiple words.
P.S. By the way, I'm not at all dissuaded by something that is inherently benign being popular among shitty people—in my opinion, all the more reason to take it away from them, by using it benignly more often than they use it pejoratively. It was successfully done with "queer", I say keep that train running!