The States (and sometimes, the Counties).
The United States is a federation of 51 mini-countries (50 states plus the District of Colombia, the seat of the Federal Government).
The US Federal (National) government manages aspects of life in America, but it delegates management of many aspects of life to the States (and the District of Colombia).
Vehicle registration (as well as vehicle purchase and vehicle insurance) are managed by the States, not the Federal Government.
This means that there isn't just one process for registering a vehicle in the United States; there are (at least*) 51 different processes.
So which state's process are you required to follow?
Each state has a law requiring residents to register their vehicle(s) in said state within a certain period of obtaining residency (usually 30 or 60 days).
What if you aren't a resident of any US state?
If you are an international visitor, not a resident of any US state, you get to choose the state in which to register your vehicle.
*While many states have a single vehicle registration authority (often called the DMV, or Department of Motor Vehicles), some states, such as Tennessee, delegate vehicle registration to the Counties, the level of jurisdiction beneath the States.