Your DNA IS your identity, and sending it to them is sending them all the information ever needed to identify you. For an encore you are also helping to identify your parents and your children and other relatives to a lesser degree depending on how far they are removed from you in the family tree.
That there is a name and a social security number attached to your DNA right when you send it is a convenience, not having that information is not an obstacle if any of your family members are also 23andme customers.
The only way this would work is if everybody sent them anonymous samples and 23andme would destroy each and every sample after receiving and processing it and destroy any and all records created as a result of processing that sample. No way they'll do that, and you could easily argue the only reason they exist is to build up that database.
I wrote about this a couple of years ago in some more detail:
http://jacquesmattheij.com/your-genetic-information-is-not-j...