It's funny how context changes everything.
Nobody would bat an eyelid if you had said, "I'm proud of being black." It would not have connotations of racial superiority. Instead it would elicit feelings of self-respect, overcoming the odds, etc.
If a person says, "I'm proud of being white" then immediately the suspicion is that that person holds feelings of racial superiority.
It's just how it is. I think it's really obvious why this is the case (once you apply context), but it's worth pointing out regardless.
I think that it's just a matter of which irrational prejudices are currently fashionable. It's always been the case that identical words/deeds/etc are treated very differently depending on which group identity they are directed against(*). All that changes is which group identity is a fashionable target. Sad, but that's how it is and always has been. The idea of equality, and I mean genuine equality, always loses to irrational prejudices.
In reality, the same position is the same position. Being proud of being the "right" race is the same as being proud of being the "right" race and it
always has connotations of belief in racial superiority because (a) it requires belief in race and (b) it requires belief that one race is better than others because that's where the pride comes from. You can't be proud of something you don't think of as being better than the alternative.
Be proud of whoever you are. The hell with everyone else.
But happening to be put into a particular group that was made up in an arbitrary fashion and isn't really a group at all in any meaningful sense and is irrelevant to almost everything and nothing you had any control over, let alone anything you achieved, is not who you are.
The idea that an obviously inaccurate description of the colour of my skin, made up to rationalise irrational prejudices, is what I am is ludicrous.
* It's always a matter of against, not for, regardless of how it's phrased (the phrasing is also just a matter of what's fashionable). Irrational prejudice in favour of group identity A is irrational prejudice against group identity not-A.