it depends on the aim
I think the cat doctor's right here, but is convoluting the 'aim' to make the law fit an agenda that is opposite to what the court actually said.
Example 1:
The Hairy Women Club focuses on the relative merits of depilation strategies, and whether it's better to remove hair or let it all flourish.
This is a topic of interest to both female people and gendery males. They form the bulk of the membership, it's all good.
Non-gendery males could also join, however, as the club is mixed-sex under the terms of the Equality Act. They can't exclude applicants because they're men.
This club can exclude people with full-body alopecia.
Example 2:
Celebrating Femininity is an association of people with a 'female' gender identity. It includes gendery males and women who, believing they have a gender identity, self-describe as cis.
Although these members share a characteristic of gender identity, I think they'd be obliged to accept other women and men as members. They can exclude people who think femininity is bullshit, due to their group's interest.
The group's common aim is clear, but 'gender identity' is not a protected characteristic: it's gender reassignment, which obviously doesn't apply to the female members. So it's a mixed-sex group for people who celebrate femininity.
The Equality Act is very clear on what's meant by 'gender reassignment'. If I (female) suddenly decide I have a feminine gender identity after all, this is not a reassignment so I can't demand inclusion in a group of trans women.
... In my view, it's insane to build a legal framework on subjective feelings and socially-constructed concepts (gender, race, religion). But the laws we have do recognise those feelings & concepts, protecting our rights to them. I understand why they do.
In order to protect these rights, they have to be well-defined: and they are. You can't just decide the EA meant something else, really, and then expect to remain unchallenged. If you want your flamin' identities to be ratified in law, campaign for it. Don't just ride roughshod over the existing legislation.
(I didn't realise I was going to write a mini-essay! Thanks for getting to the end.)