Common definitions/understandings are the only way we can make sense of the world. We have to agree on what a term means in order to use the term for any purpose. I can not ask you pass me the glormph, because that term does not communicate anything useful until we both define it in some similar fashion.
So sure: identity can be defined by the individual for the individual's use. And nobody else can experience or understand their identity, let alone define it, and therefore nobody can tell them it's wrong. Fine. I think it's a dumb point, and shallow thinking, but fine.
Except that's NOT where it ends, for many trans people. For trans rights advocates, the rest of the world must affirm that individual's identity and his/her definitions. The rest of the word must change their own use of language to match the trans person's meaningless definitions (ie: our definition of 'woman' must now include all those who simply claim they are one) ; the rest of the world must redefine 'man' to mean what trans people want/feel, instead of the historic 'adult human male' definition. If I refuse to verbally acknowledge these new, meaningless definitions and uses of the language, then I can be charged with a crime and be fined or go to jail.
Nowhere else are these demands permitted. A Muslim may have a spiritual identity, and define himself through and by that, and experience it, and it is 100% reality to him. But that Muslim does not, legally, get to go around and demand that everyone else respond 'Peace be upon him' every time he utters the name of Muhammed, or that we must pray with him 5 times a day. The Muslim does not get to cry 'You're denying my right to exist!' when an atheist says 'I don't believe in God' or when the school curriculum teaches evolution. It's not a hate crime to refuse to go along with what a Muslim/Christian/Buddhist claims to be true.
For decades, Trans people lived in a similar way. They made the best choices for themselves, based on what they experienced and felt, but they did not expect society to align itself with their worldview so that they could avoid having their feelings hurt.
If you are demanding that I accept a definition/term/understanding, then you'd better be prepared to define it clearly. If it's illegal to NOT accept that definition/term/understanding (which it is in Canada and the UK), then the definitions had damned well better be clear.