You linked this in response to my own piece on Jordan Peterson's suspension from Twitter, so I'm going to react here.
Your definition of identity seems incorrect, though I understand the overall point you're making. People's identities change in all sorts of ways, which means that it isn't "a property that doesn't change from one observation to the next". Identities change as we age, learn new skills, get univeristy degrees, get new jobs, change our bodies, get married, etc. etc.
However, I agree that when people talk about "transitioning" in the sense of a trans person living as the gender opposite their biological sex when they previously were living as the same gender as their biological sex, they have changed their identity (note: the word trans is not a prefix that means "transitioning", it merely denotes how a person's biological sex is opposite their gender identity. See the scientific terms "trans fats" and "cis fats" for a good illustration).
In that sense, it can seem odd for someone like Caitlyn Jenner to say "I've been a woman all my life". But really, all she would mean by saying that is that her gender identity was always that of a woman. It's just that she lived as a man prior to transitioning. So when she's referring to "woman" she's really talking about gender identity, not biological sex. That means someone could be trans without any physical or gender expression hinting at the fact they really identify with the opposite gender.
Does that mean anyone who has a fully male appearance and fully male gender expression should be accepted as a trans woman by mere declaration? I would say no, at least not without something more to back it up. If this hypothetical person started taking male hormone blockers and female hormones, or at least dressing and behaving in ways that signal they belong to the opposite gender of their birth sex, then I would stop being so skeptical.
Whether we should retroactively change the name of someone who adopted a new name when they made their gender transition, and refer to them by their new pronouns when discussing them in their pre-transition past, is an interesting question. I don't really know the answer, but I agree it doesn't seem like the worst transgression (see what I did there?) to deadname a trans person when referring to them before they transitioned. If Elliot Page was always Elliot page, then the phrase "Elliot Page used to be Ellen Page" becomes false, and yet on its face it seems manifestly true.
Finally, I think a soul would probably be a good analogy or metaphor for what gender identity is. Having a gender identity that doesn't match one's biological sex is not something I can personally understand, nor most people, but its existence as a persistent phenomenon across cultures and time prove that it indeed exists. There's nothing empirically verifiable, like a certain brain module, that can tell us that someone's gender identity is the opposite of the biological sex. At least, not yet. The same is true for homosexuality though, and no one would seriously deny its existence. In fact, the same is true for many purely mental phenomena, like pain. The only manifestations of these brain states that we can observe are behavioural. That doesn't mean they don't exist.
