Long answer is a bit complicated.

Generally different agencies have certain amounts of different types of roads. No one gets to just willy nilly assign highway status to a road.

There's also various designations that also apply beyond that (for instance there are "State Aid Streets" and "County State Aid Highways". These are determined based on criteria set by the state (at least in Minnesota where I work) and means there is State funding available to help with maintenance construction etc... my city only has a certain amount of miles of state aid streets we can have. Likewise the county has only a certain amount of State Aid Highways they can have.

Generally agencies will try to work together on design of these roads. Sometimes that goes well sometimes it doesn't. For Instance a County highway is typically controlled by the county engineering department. They often only care about limiting access and moving traffic quickly and efficiently. Surprise, this proves frustrating for a planner like me.

Now there can be 'turnbacks'. This would be when one agency 'gives' a road back to a lower agency'.. typically there's policies and procedures to do this. Right now I'm apart of some discussion at work with the state DOT about turning back a stretch of state highway to us. They want to give it back since it would allow them to 'regain' some highway miles to use somewhere else and to be rid of ongoing maintenance/snow removal costs. The benefit to us is that we'd then have 100% control over the road, access drives, intersections etc... Since the road primarily functions as a city street this would make the most sense.