Funny you ask.
In Florida, you don't actually require a fixed address for registration. If you're homeless, you can register you car....and list the place it's most frequently parked as your residence/address.
But..... normal registration requires ownership/residence in the state unless affiliated with military or other specific reasons.
Virginia would/will be quite interested in how one of it's residents does not have a complete tax profile for state/local/personal property/real estate taxes.
Of course, Florida might well wonder the same thing.
Cross-registered homes/autos/property/income/residence taxes and registrations etc... will eventually catch up with them if they're not careful and not filing partial-year returns. The fees, fines and late charges far-outweigh any possible benefit. Fairfax, Arlington, Falls Church etc... all have 'out of state' license programs targeting those residing in county without a Virginia license. Ditto for DC and MD. Ever heard of a 'plate reader?'
So, the answer is NO: you can not legally pick and choose where you would like to register your car, your state residency etc... based on a whim.