That, or, establish a trust to own the vehicle and grant yourself permission to use it. It's not exactly trivial to do and costs some money, but it's doable.
You can do similar with an LLC, but that gets more complicated with the rules regarding using a "company" vehicle for personal purposes. IANAL
Similar things are done for things like cellphone plans, firearm ownership, homes, etc.
The only thing I am aware of that you can only do in your own name is register to vote. Almost all of the Michigan voter database can be FOIA'd. It's called the QVF - qualified voter file. Only a few fields in the database (ie, day and month of birth) as well as all voter records for victoms domestic battery are protected by statute.
Pay a thousand bucks to a Montana agent to register you an pseudoanonymous LLC and put an old 90s Corolla into an LLC with a permanent registration plate (since anything 15+ years old can have a 'perm' plate on it.
While some people do get away with this, it carries some risk.
Without using an LLC, most every state requires you to register your vehicle where you live within 30-90 days with some exceptions (ie college students).
Even with the LLC, if you catch the attention of the state, I believe you might be risking being charged with tax evasion even if your goal was to protect your privacy. This is especially true if you can't prove the LLC to be a legitimate business venture.
Yeah, the Corolla won't be mistaken for a supercar, but many states have begun cracking down on residents with Montana plates such as Georgia, Ohio, and New York.
Also, insuring a car with out of state registration can be committing insurance fraud. Rates and fees are different between states due to different regulations. Further, depending on your policy, the insurer could deny claims because the car wasn't garaged in the state it was registered.
Really, if the privacy is of sufficient priority, the best solution is to just do things properly and move to rural Montana instead.
Permanent rental it is then. :)