The short story is yes generally all your stuff should be registered in the state you claim "residency" in. (Not to be confused with your tax home which typically is the same but can be different for various reasons.)
Residency, registration, licensing or licenses though can be very subjective depending on the personal situation.
Having moved 9 times now in the last 20 years, different state each move, I can say through experience that for the most part it doesn't matter as long as everything is up to date on the tags.
Have had enough speeding tickets etc where when I get pulled over my drivers license, plates and sometimes insurance show up to 3 different addresses in 3 different states and I have not been hassled or cited for it.
You'll find some cops that want to use it as a revenue generator if they see this, but if there's a halfway plausible explanation then it shouldn't be an issue.
If you read some of the registration/ insurance requirements, they could have a person changing plates and drivers licenses at a rapid pace based on location.
IMO find something else to worry about.
