The very foundation of the alt right movement is rooted in this very belief. Hence, the main reason for this posting.
Why didn't you specify alt right in OP?
Given how nationalism works (in theory), I can see white nationalism as being inherently prejudicial likely leading to superiority claims of white people that truly support it. Same would be true with black nationalism or gay nationalism, etc. But with pride, alt right doesn't alone get to claim ownership of such pride. They can try, but they'd be mistaken.
I agree with
@suncowiam in that there is double standard at work here, and it is plainly obvious. Pride in and of itself is not the problem. To become a problem, it needs the explicit claims of superiority being mentioned. I think current white nationalists have no problem (really none at all) in laying claim to such superiority. But white pride in and of itself, and apart from the connotations would not be inherently prejudicial, nor would be exclusively discriminatory toward whites only. Again, plausible to have pride in more than one thing.
If somehow this is not sinking in for some on this thread, then I would think pride is the problem. I don't think it is that, but perhaps definition of "pride" would help. Notions of some things are inherently better to have pride in than other things are opinions, and borderline superiority complex at work.