Right, but couldn't you just give them a fake name and use a PO box, or am I missing something?

Sure, but you would be giving them your DNA, which can be mapped back to you through other peoples DNA that are in any way related to you. What I am saying is, that once someone else has your DNA, all aspects of anonymity are gone.

To reduce exposing "who" you are, then you would have to ensure that nobody currently related to you, or future offspring, ever get their DNA stored, are never arrested, are never in a medical facility that stores DNA samples, and so on. You would also want to verify that nobody related to you has ever been arrested or provided their DNA willingly or otherwise.

DNA can also be used in conjunction with computer modeling to draw a picture of approximately what you look like. That image can be compared to state ID card data. This was done for a brief period in Japan, using chewing gum stuck to bus stop benches, to show off their software. Here [1] is one example from 2013.

In summary, to remain anonymous, you would need a kit that you operate at home, that does 100% of the diagnosis and reporting in your home, with no DNA and no data leaving your home whatsoever.

[1] - https://www.smithsonianmag.com/science-nature/creepy-or-cool...

If your brother or sister submits their DNA it's going to be immediately obvious you are siblings. Same goes for your parents. There is some ambiguity if it's aunts or cousins, but you might be able to work it out.