You don't get that a lot where I live, but I have yet to meet a nanny here (people are more likely to send their kids to daycare or stay home), and anyway Vancouver has a large variety of ethnicities. I think caucasian makes up the largest fraction, but barely -- maybe half? (And it's not necessarily the wealthiest fraction, either. When Hong Kong got handed back to China in 1992, huge numbers of their very affluent residents immigrated to Vancouver, bringing money with them).
However, we used to live in Marin County (just outside San Fran), where the population could be neatly divided into "white" or "Latino", and I was horrified by the way that the Latino population was viewed as basically second-class. Many were illegal Mexican immigrants, so maybe that was part of it, but I remember how if you passed a Latino on the street, he would quickly avert his eyes and look down, as if hoping not to be noticed. Anyway, I can remember many a day at the park with my blond, blue-eyed, fair-skinned daughter and a whole horde of other blond, fair-skinned toddlers and I was literally the ONLY non-Latino adult there. I met a lot of lovely Latina nannies, and I can certainly imagine how a middle-class Latina mom could easily be mistaken for a nanny there!
So, to answer your question -- unfortunately, I think it is reality, and yes, it's also offensive. I won't ever be in that position myself (my husband and I both have the lightest of light blue eyes and fair skin, plus the kids look like a computer-generated morph of the two of us -- we're clearly a family), but I can sympathize with the writer's frustration. I think she and the other dad handled it well. We need to teach kids not to make assumptions.