That's a good point. But prior to empire and the subsequent arrival of immigrants people in England, the main signifiers were class and not race. I would wager it was the same across Europe. When everyone is white no need to identify as white. Humans being humans will find a different way of dividing themselves and identifying.
It's a good intention but to be honest I don't see it holding over the long term unless there are other changes e.g more mixed urbanisation. We tried the same thing under Siad Barre where mentioning tribe was banned (IIRC) but as we saw it never went away. Saudi Arabia has also banned tribal websites and tribal flags on cars etc in attempt to build a ''Saudi'' identity and prevent power bases separate from the Al Saud. But people are still
as the surnames are a tribal label.
Banning tribal labels seems to me a high minded ideal pursued by authoritarian regimes that usually ends with the fall of the regime.