The far-right tends to believe that girls should play with dolls, that gender roles are tied to biology and inherently unavoidable.
The gender identitarians believe that if a child plays with dolls, the child might be a boy, and that gender roles are not tied to biology, but inherently unavoidable. So being a woman means accepting those gender roles, just as it does in the far-right theology.
The difference is that anyone can choose to be this new concept of 'woman', as long as the person agrees to behave according to sexist female stereotypes, whereas those stereotypes are tied to biology in the far-right framework.
The far-right also believes that only men do heroic things in the public sector. The gender identity ideology posthumously transes famous women in history, because it seems that only men do heroic things in the public sector.
Those are some of the ways the horseshoe theory applies to this question in politics. Both are also deeply sexist, and this can be seen among the far right in how its decision power is clearly held by men in public spaces, in gender identity theology this can be seen currently in how only the female sex is being erased (we are now egg producers, vulva havers etc.), while the male sex is carefully left alone (no sperm producers, ejaculators etc.) and still called men and boys.
The gender critical approach does share something with the far-right views, and that has to do with the biology-based (lived experience) definition of 'woman.' But the goals of the far right are diametrically opposed to the goals of gender critical feminism:
The former need to have a clear name for the female sex in order to subjugate and control it (and its reproductive, sexual, and support-role resources), the latter need that clear name to fight against those exact attempts.
So yes, overall the gender identity ideology is a giant leap backwards in women's rights. It tries to return to 1950s gender roles (perhaps because if those rule then a man can transition by simply following all those rigid rules, while if they do not rule, then transitioning is harder as everyone ends up gender-nonconforming).