That's not exactly correct. Driving is a privilege, but voting is a legal right defined both at the national and state levels.

You do have to prove you meet the eligibility requirements to possess that right, but having done that, you can not be denied that right unless you forfeit it, such as by committing a felony. The privilege of driving, in contrast, you not only have to prove eligibility, but also competency (well, for certain, very lax definitions of competency), and are subject to ongoing regulation of your behavior while driving. If your driving privileges are revoked or suspended, you have more limited legal recourse for restoration than you do for your right to vote.

I think you are right to raise concerns about potential for conflating the two if people and even political leaders start to associate voting rights with driving privileges, but I don't think that inherently makes automatic registration a bad policy.