So if you and your identical twin brother or sister are both going to apply to 23andme for an analysis you should get back the same results, and if your identical twin brother or sister commits a murder and hightails it to zanzibar you are in a lot of trouble if their dna is found at the crimescene and you happened to be in the neighbourhood.
The number of identical twins is low enough that for the larger part of this discussion you can leave them out of the equation, they're the exception that confirms the rule, the differences in DNA are very subtle, and afaik will not be readily detected in a normal analysis.
Random link of Google: http://www.nytimes.com/2008/03/11/health/11real.html
Of course they are not, but for 99.6% of humanity the rule that your DNA is (very) unique holds true, and for identical twins even if their DNA is even more similar it is not 100% the same.
Clones (which do not exist (yet, or at least, human clones)) are the other exception.
All you're missing is the metadata of 'name, date of birth, social security number' and so on. But that's just a system we've placed on top of 'who you are', which is your genes.
Your fingerprint, your retina scan, your full lifes history, nothing in that list will identify you as good as your DNA does.
So once you have someones DNA you can reconnect it with the rest of the information at any point in the future by simply checking to see where that DNA turns up again, including fractions of it in your descendants.
You can't really anonymize it.