Not give information out. Keep it to yourself. With or without technological aid.
The individual trying this out either has to lie a lot (give false information) or risk standing out. That is, in the current environment.
To make this possible we need our systems (in the broadest sense of the word, so a government keeping records is a "system") to change the working assumptions. For many things you still "need" to give your name when in fact there is no need intrinsic to the function, only convenience, usually for somebody else, like law enforcement. This "convenience" carries risks itself, so it is long term all around better to get rid of it.
How you'd do this? Well, change the assumptions, change the laws, change how we organise things. Only after that does technology come into play.
And that technology? Authentication systems that aren't inevitably tied to just one possible identity per person and certainly not systems depending on some selection out of a small number of possible passwords per person with no redress possible, and also not systems that depend on a limited subset of "identity providers", reducing everyone else in the system to second class citizens.
Come up with designs that address those, and more shortcomings, that empower instead of subjugate. You might even consider deploying DRM around packed-up identity information, giving control back to the owner. Better yet, don't give the information away at all, for example using zero knowledge proofs. There are plenty of tricks ("technologies") and we are employing far too few of them.
Even the simple act of not requiring everyone to use the same database key for completely unrelated databases is, so far, too hard, and that needs to change. I could go on, but this ought to suffice for now.
While there is no single silver bullet, there is a clear general direction how we need to change our current systems, how we need to change our very thoughts and assumptions to keep a tenable society. Notice that everything so far governments and corporations have produced and are producing, fail at the very first of assumptions. A good case in point is "NSTIC", who were warned of these issues yet, as is evident in the design, chose to ignore the warnings.