Not everything is binary. Driving is a privilege, yes, but making use of the privilege does not imply forfeiture of civil rights. A balance has to be struck.
And many, many people are perfectly ok with the balance of "if you want to drive on public roads, you have to display a somewhat-opaque identifier on your vehicle so you can be held accountable if you screw up".
As people point out, the opacity has lessened over time, and it's reasonable to question if we should re-evaluate how it's being used. Kind of like how social security numbers have changed their role in society.
Being required to identify yourself and related property in relation to suspected crimes (with probable cause) is acceptable. E.G. "License and Registration please, you are suspected of committing X and possibly other crimes."
Being required to wear a real-name badge in a public place is much more unsettling.
License plates not being required will lead to abuse of driving privileges, an imbalance.