Idk, it probably has some value. But my point was that it's going to be difficult to prevent your real identity from becoming attached to your DNA forever. The moment your real (DNA, identity) pair leaks from a credible source, your privacy is permanently and retroactively ruined.
So if 23andMe leaked a fake name with your DNA, it's out there in the hands of advertisers/scammers/governments/etc. From now on, anyone who gets access to your DNA will be able to build up data on you, and all it will take is a single leak/sale from a credible source to make it accurate.
(...but in truth, I have no idea what "DNA data" looks like, or if it's even possible to use it for targeting...)