I don't know what started this idea, but everyone says it and I just don't get it. It's explicitly stated that Stormcloaks don't hate anyone who isn't a Nord. They may use the term "True Nord" often, but "True Nord" doesn't necessarily mean you have to be a Nord, you just have to represent the ideals of Nordic culture, which is a role than can be fulfilled by any race.
All Dunmer aren't the same, so a Dunmer could join either, but it's more likely that a Dunmer would join the Empire as opposed to the Stormcloaks. Hell, you can even find a Dunmer farmer on the road occasionally, claiming he's off to join the Empire.
"True brothers and sisters of Skyrim" does not include any of the interloper races... ask them. Better, run into them on the road. They call everyone a milk drinker except Nords.
Hate is a strong word... unfair. I'd agree.
How about this? The Stormcloak Nords are radically closed minded and unable to see the big picture that includes all the races of Skyrim working together. They've been indoctrinated by extremists and like all extremists they have a very narrow view of just about everything because if they opened their minds they'd have to accept that what they're doing and saying and believing seems pretty hollow and narrow in the scheme of things. If they didn't have their backs against the wall and the rumors of you being a Dragonborn weren't fresh in their minds, I doubt anyone but a true Nord would be able to become a Stormcloak.
Also, a point to ponder... we are actually in the lands of the Snow Elves, a grand and beautiful race of Elves who are all but extinct and forever changed by a combination of Nord occupation and forced slavery by the Dwemmer. Anyone else came later and ultimately took what they got without thought to the natives. The true brothers and sisters of Skyrim live in caves without eyes and make creepy noises when they come to kill you.
lol