Yes for various periods of time there were various efforts to construct such an identity, to serve various ends of the people doing so. This kind of process has taken place all over the world, throughout history, in various societies as well.

In the case of "white", "black", "brown" and so on, those definitions also changed considerably over time. Then, at some point, the efforts to intentionally draw hard lines around petered off and lost support.

Those labels are still used because they do have some kind of descriptive capacity, even if solely because of momentum from old habits.

Crucially: none of this is responsive to my point. There's no good reason, today, to assign "white" to mean "prejudice + institutional power" in a general sense. It's not necessary to do so. Doing so is not sufficient to address and unpack all the dimensions of prejudice and institutional inequality as they take place today. But doing so does incur all the imprecision-based racist harms that I described in my post.