A law that can only be sustained through the deliberate avoidance of proper definitions is an unjust law.
If, by the simple act of fairly and accurately defining key terms, the law would collapse, then the law SHOULD COLLAPSE. If the only thing holding a law together is avoidance, obfuscation and circular fallacies, the law isn't fit to stand.
"Physiological and other attributes of sex" is as demonstrably stupid as "physiological and other attributes of a pancreas".
A pancreas is physiological.
Sex is physiological.
There is no "other" attribute of sex, that is not physiological.
Prettiness? Fragrance? Musicality? Clothing? Language? Names? None are attributes of sex.
A physiological state is a physiological state, and just as any and all conceivable alterations to the physiological organ 'pancreas' do not change the pancreas into a kneecap, nor take it one iota closer in definition to a kneecap, similarly neither do any and all conceivable alterations to the attributes of 'sex' change it in any way. It's impossible to change sex.
A law that 'recognises' a pancreas as a kneecap because of 'physiological or other' changes is no less preposterous than a law which recognises the male sex as female because of 'physiological or other' changes.
Almost two decades of forcibly trying to legislate the physiological states of male and female into a fictional state of 'psyche' has failed, utterly.
It's been a miserable experience for all of us still living in the physiological reality of our female sex to endure this failed experiment.
So to answer your question, OP, 'gender identity' is the lie that states that we all have a gendered psyche, and it is NOT enshrined in law.
And 'gender reassignment' is the legal fiction - which is a delightful euphemism for 'big fat lie' - is the lie that there are "physiological and other" attributes of sex that can be changed in any meaningful way. And that IS enshrined in law.
For now.