If we want them to be uniquely identifiable, then we don’t need to hide the identifier. If we don’t want them to be uniquely identifiable, then hiding the identifiable ID becomes a violation of the voters privacy.
Also IANAL but I feel like there must be some law about deception of this kind. Selling a product that does something you don't expect it to, were never told about, and would not reasonably even observe feels somehow like fraud.
Or you can register as an independent if you prefer. The rules vary by state, but the worst that happens is you can't participate in the primaries (but can absolutely vote in the final election). Who you, as an individual, voted for is still not traceable back to you.
In other words, it's your right, thanks to the secret ballot, to vote completely differently from the party you registered under or even differently from how everyone in your life perceives you, and nobody will ever know.
There are people who changed their mind years ago about what party they support and have been voting that way ever since, but just haven't gotten around to (or don't care enough to) update the party they listed on their voter registration.