This in effect makes my DNA a matter of public record based on the decisions of my relatives (behind some hoops, but in practice it doesn’t seem hard). That seems like a breach of privacy.

Well it's technically your uncle's DNA. Limiting what he can or can't do with it because it also matches yours would then infringe on his rights.

Sure, in the same way my neighbors infringe on my rights to blast music at 4am.

Not the same. Your uncles DNA only establishes he is your uncle. Unless your also saying that your uncle is not free to disclose any information that might be contained in his DNA. For example that he is a white male.

You dont own your relative’s genetics. They can do whatever they want with them. There’s no breach of your privacy because nothing of yours was breached.

But my dna can now be matched to me - That is a breach of privacy today, or at least and end run around legislation that attempts to limit the spread of genetic information.

We should think really carefully about this! My uncle’s right to check out who his long lost cousins could put my genetic information in the hands of a potential employer, insurer, or government. That could be really bad.

This is similar to the Facebook scenario, where it’s not just your privacy you are trading away for whatever benefit, but those of people related or near you.

Are you saying that your uncle also can't post his full medical and psychological history online either?

Not the OP, but until not that long ago there was a social pressure thing of not "advertising" your relatives' history of psychological problems (mostly suicide or time spent inside a psychiatric ward) to outside members of your family, lest those outside members would think you had the potential of having the same issues, too. This is sort of similar to the "uncle" from your example not being free to post "his full medical and psychological history online", I mean, had he done that his family would have certainly retaliated by telling him to remove that information from public knowledge.

Btw, I live in Eastern Europe so I'm talking about stuff I've experienced here, maybe in the States or in Western Europe this behavior changed a long time ago and not only recently.

There's a big difference between your own DNA being publicly hosted and identified, and some DNA that is pretty similar to yours but not the same as yours being posted and identified to someone that is not you.

Why should you have a right to block the later from happening? It's not your own personal information.

I think the difference is much smaller than you think. If someone finds male DNA and my uncle has used this service, and a public photo with my eyes and hair clearly visible in the background on his Facebook page, then suddenly it’s almost as if I registered my DNA in a public registry. It’s the metadata problem all over again, it just takes one or two more public data points to deanonymize it entirely.

since DNA is by nature unalterable, that implies that there is no right to privacy of DNA.