According to this article, gender identity might de facto replace the sex at birth based on responder choice:
Scotland’s chief statistician recently issued guidance stating that data on sex should only be collected in exceptional circumstances.
The Office for National Statistics (ONS) was also embroiled in controversy when it proposed to guide respondents to the 2021 England and Wales census that they may answer the sex question in terms of their subjective gender identity, rather than their sex.
The same article mentions that quantitative social scientists are troubled by such a change:
(quantitative social scientists) who believe that data on sex is vitally important and that data on both gender identity and sex is needed
I have tried finding similar examples outside of the UK and the closest I could find are in Canada, where the public awareness and sensitivity towards LGBTQ2 communities has grown substantially. However, they seem to favor having both "sex at birth" and "gender identity" related questions:
What was this person’s sex at birth? and "What is this person’s gender?"