Yep. Donair and gyro meat are pretty much the same thing. Donair is probably a corruption of Döner, which is the Turkish name for a Greek gyro. (or maybe gyro is the Greek name for Turkish Döner, who knows)
The traditional sauce is Tzatziki, though: plain yogurt (preferably whole-milk), fincely diced garlic, diced cucumber, salt, and dill. I've never heard of the vinegar-milk-sugar sauce listed above, and my sweetie, who spent quite a bit of time in Turkey, said "they put WHAT on their Döner?!?!" when I described it to her.