A person is adopted. Who are their real parents?
Some would identify their adoptive parents as their real parents, especially if they have the good fortune to have a loving and supportive adoptive family. Others would identify their birth parents as their real parents, especially if they dislike, distrust, or resent their adoptive family.
It is generally understood that the correct answer is the one chosen by the adopted person. To tell them that their real parents are the ones they didn’t choose – that would be beyond rude.
Therefore, it is (and should be) a social rule that one’s real parents are defined by self-identification.
Transgender politics can be understood as promoting a social norm of adoptive gender. A person’s birth gender may differ from the gender they chose to be a part of, and just as society supports and acknowledges family transition (adoption), so also trans advocates call for society to support and acknowledge gender transition. Just as a person may declare their adoptive family to be their real family, so also they may declare their adoptive gender to be their real gender.
(It is rare for a person to declare their adoptive gender to not be their real gender, because coercive gender transition is rare. Adopted people who reject their adoptive family are usually not adopted with their consent. However, coercive sex/gender [re]assignment does occur when an infant presents as intersex, and many intersex people find that they disidentify with the gender assigned to them.)
This framework also provides a way to judge the validity of transracial identity claims. An adoptive family derives its validity from the consent of all its members; if Alice identifies Bob and Carol as her real parents, but Bob and Carol do not agree, then Alice’s claim is suspect. Similarly, the validity of transgender identity claims depends on the support of cisgender advocates of the adopted gender, who welcome and acknowledge the transitioner as a full member of their gender. In this framework, then, the validity of transracial identity is to be judged by the existing racial community of the adopted race.
This is also how religious conversion works. The rules for who gets to call themselves Jewish are defined by the Jews. Likewise Hinduism is defined by the Hindus, Islam by the Muslims, Buddhism by the Buddhists, Christianity by the Christians, and so forth.
In full generality, membership in a community is defined by (1) the existing members of that community, and (2) the individual who is to be considered a member. If either of those sources dispute the membership claim, then the membership claim is most likely invalid.
(Note: this section deals with details of religious beliefs that I do not personally share. If I have misrepresented these beliefs, then I apologize and would welcome any corrections you see fit to send me.)
There is another parallel between gender transition and faith: transubstantiation. During the Eucharist, the bread and wine are held to become, respectively, the flesh and blood of Christ. It is generally acknowledged that the physical properties remain in place – the color, smell, taste, texture, nutritional content, and so forth are as one would expect from bread and wine – but these materials, once consecrated, nevertheless constitute Christ’s flesh and blood.
This strikes me as being very similar to the way in which (for example) the body of a pre-transition trans woman may be medically indistinguishable from the body of a (cis) man, but due to the choice and intent of transition, that arrangement of matter nevertheless constitutes the body of a woman. A trans person’s body is transubstantiated into the body of a person of their correct gender; the Eucharist transitions into the flesh and blood of Christ.
The common thread here might be cynically described as wanting to believe in something regardless of its factual basis, and being willing to fudge definitions and ignore facts in order to make that happen. But that definition applies equally to adoption – the choice to identify one’s adoptive parents as one’s real parents results in no medically detectable change, no rewriting of genetic markers – and yet adoption is widely regarded as legitimate. So the process of redefinition by will has clear precedent, and is capable of being valid. It remains only to determine whose consent is required for the redefinition of which terms.