This was an issue at my last employer. We would send out surveys, one of the questions regarded race. We had started out using the ONS classification, but this became a problem as soon as we got US customers. It became even more of an issue when we started dealing with multi-national companies.
In the end we changed all those questions to something along the lines of "Do you regard yourself as a member of a racial minority?" (not exact wording). Not only did it avoid the classification issue, but it actually focused on the question that we were trying to answer, which was whether our customers had a problem with racial bias in their hiring processes. The actual "race" of the candidate was irrelevant at the level we were addressing.
It was a great example of software that asks one question in order to answer a different one (like asking for gender when you actually just want to know how to address your emails to the customer).
Adam Rutherford (geneticist and broadcaster) has had some great rants in the past about the falacy of "race" as a scientific idea, and how the boundaries between supposed races are social constructs overlaid onto group differences. He has a great/depressing three-part radio documentary about it and the rise of eugenics called "Bad Blood".