In every state that I have lived in (which is 7), there was a requirement of registering vehicle and getting drivers license within 30 days of arrival, assuming you were in the state with the intention of residency (i.e. not just a short visit)..
Of course, the devil is in the details. And honestly, they don’t know if it is 30 days or 60 days from when you moved to the state, so you can get some leeway.And as others have answered, students and the military have a different set of rules.
But assuming you are a non military adult and you have moved from state A to state B, you have around 30 days to retitle your car in the new state, get auto insurance that is applicable in the new state, and get license plates.
You might get by with not doing it, but the problem is if you are pulled over by a police officer and you have a Florida license plate (for example) and an Illinois drivers license. You’ll get a ticket.