They pretty much do.... they allow dunmer in windhelm, but they have to live in the slums, argonians and kahjiit aren't allowed in the cities, the only exception is Riften, but Riften is often referred to as a corrupt impoverished slum.
The Dunmer were given that part of the city after they fled Morrowind almost 200 years ago. They were immigrants seeking refuge in Skyrim after catastrophic events in the homeland, and were given places to live under their own sovereignty within Skyrim's borders where they do not swear fealty to local authority or pay taxes to support public services. They are supposed to be self-supporting and self-governing.
Neither Ulfric nor the Stormcloaks put them in the Grey Quarter, they've been there for several Nord generations. Any previous residents - some of them, no doubt, tax-paying citizens - would've had to leave and find other housing when the Dunmer arrived. It was a gift to an influx of otherwise homeless refugees who wanted and were allowed to remain a separate people within the city, hold, and province, responsible for their own affairs. We don't know whether it was a "slum" or not when they moved in there. If it wasn't, then they have failed to keep it from becoming one; if it was, then they have failed to do anything about it. They are not paying any rent or any taxes, which is why they are not being extended the same public services. The money they are not paying in rent and taxes is supposed to be theirs to do with as they please. If they're not using it to provide public services within their own community that isn't Ulfric's fault.
As for the Khajiit, they are not allowed in any cities, even the ones controlled by the Empire - before, during or after the war and regardless of its outcome. Using their situation as a slam on the Stormcloaks is a failed argument because the Empire doesn't treat them any differently.
And FWIW there will be no changes of note in Windhelm after the Empire's man replaces Ulfric as Jarl and the Legion controls the city. The Dunmer will still be relegated to the Grey Quarter, the Argonians will still be relegated to the docks (that's actually the new Jarl's stated policy) and of course the Khajiit will still be kept outside the city walls as well.