Can you imagine if this was 'race ideology'?
People saying they identify as (some race) - citing how they always really liked the (racial stereotypes) and felt wrong the way they were, their own racial stereotypes did not resonate, so they now know that they really are the new race. They demand that members of that community validate them and refer to them like they were never any other race, like there was literally no difference, like it was literal violence to discuss skin tone or ancestry or upbringing because they are the most vulnerable, those who are black inside but everyone thinks they are white (or whatever). Everyone else has to love their stereotypes and if they don't want to switch to another race they are cis-racial and they are all privileged because they all identify with these stereotypes.
The biological basis for sex is far, far stronger than the biological basis for race (which I gather is actually pretty weak).
Yet I can't identify as American or Chinese and get my citizenship and validation; I can't identify as indigenous or black and make indigenous of black communities legally accept me as a tribal member or as eligible for awards for black business people. (Nor should I be able to!!).
We can't define a group according to the stereotypes assigned to that group.
So -- gender ideology is to sex as racism is to race. (not completely, not exactly - but there is truth in it).
The analogy breaks down of course, because even though racism is really bad and chronic and awful, we don't need to separate races for safety. Men are the only biologically-defined group that consistently, statistically, globally makes life more dangerous for another large, biologically-defined group -- women. Not all men, but men.