No, I don't think it is the same at all.
A British-Asian is British. The subcategory of Asian doesn't add anything. Lots of us may have international heritage if we check our ancestry but we still call ourselves British by virtue of birth. British is British.
In my example, the French person IS French. They can call themselves British because that's up to them, but they are still French - regardless of how they feel.
A trans woman is biologically male. There is no part of them which is biologically female. Referring back to your British-Asian example, the British-Asian person IS still British, even if you want to add the Asian descriptor. A trans woman is not a biological woman. Not at all. It's not an added descriptor. It's a French person who wants to call themselves British when they're not.
Unfortunately for those who prefer to solely use gender, biology matters. It is only men who rape. The overwhelming majority of sexual and violent assaults are carried out by biological men, regardless of how they identify. Men are bigger, stronger and more powerful when compared pound for pound to their female counterparts. That's why biological men can't enter most women's sports.
I'm not advocating for this person for First Minister. I'm not Scottish and tbh, I don't know much about her at all. But I do think the world would be a much better place if politicians were more honest and clearer. I think her response is excellent actually - trans women are biological men but she respects their right to identify as a woman. Pretty much covers all bases I think - other than those who are demanding that she pretends that a biological man IS actually a woman. Which is untrue and just ridiculous - but that's where we're at now and why the SNP is being laughed at.
I also very much disagree with you that the world is "increasingly" recognising gender over biology. If anything it appears to be reverting to biology. Thank goodness.