Yes, you've put your finger on the problem exactly. A person can have a stable "identity" based on false beliefs - such as that a transsexual male or a male of altered appearance & hormone profile is "a type of woman." But as you say, in most cases, it won't work when it meets reality. Since reality is all around us & tends to be hard to avoid, this means the supposedly stable "identity" really ends up not being very stable after all.

I imagine maintaining a personal "identity" based on an inherently false belief - "as a male who has done XYZ, I've now become a woman" or "any male who identifies as a woman is a woman" - must be very draining. Coz it would seem to require constantly having to filter out reality & defending oneself against basic facts. Like the dictionary definition of woman, the meaning of female, & the realities of girls & women's lives.

Seems to me that a principal reason many trans-identified people are able to hold so firmly to, & be so assertive of, their opposite-sex "identities" is because they suffer from narcissistic personality disorder or other cluster B disorders, or at the very least have strongly pronounced narcissistic traits.