I don't believe the ad was racist - to be so, it would have to intentionally demean someone on the basis of their skin colour and ethnicity and clearly that was not the intention here.
It's pretty hard to find anything original to say about soap, so the notion that it is for everyone is a good idea in principle, it was simply not through through in the execution.
It happens from time to time in ads - MacDonald's had to whip off their ad campaign showing a young lad from a broken home in a 'MacDonald's = happiness' scenario, which clearly failed to hit the mark in a similar fashion.
I conclude that the advertisers explained their thinking when the ad was offered to Dove, who figured that the public would work out the message for themselves, and in this, they were clearly mistaken.