Only khajiit caravanners aren't aloud in the cities but others are. And there are a few people in Windhelm that act hostile towards you if your a dunmer but not everyone.
Name one Khajiit in a city.
J'Zargo is it, if you count the College as part of a city. Kinda easy to see why he's the exception, since Winterhold lacks walls, the entrance to the college is right there and they definitely lack prejudices of the rest of Skyrim.
What I don't understand is, EVERYBODY hates the Thalmor. EVERYBODY. But who gets banned in Windhelm? Argonians. Who gets harassed? Dark Elves. Who's nationally banned from entering the cities? Khajiit. The Altmer? No problem, let's let them live in Windhelm, Solitude, Winterhold, Falkreath etc and never think twice about it.
It's that kind of inconsistency I don't understand and can't forgive. It's INCREDIBLY bad writing. As a real world comparison, this would be as if Palestine called Israeli jews an enemy of the state, but then had no problem letting them live in Palestine and never thought twice about it. Meanwhile, they aggressively drive out russians, Chinese and Nigerians, and label the chinese as spies. It just makes no friggin' sense.