I assume this is the road(s) along the Mississippi. Like most named highways it's mainly a marketing idea (supporting local tourism), not a formal state highway designation. It's basically any more-or-less through road that gets close to the river, regardless of whether you can see the river or not.
It might help if you specified which states you intend to cross. Travel in Minnesota will be quite different from Illinois or Louisiana.
Even in Minnesota conditions vary from US61 in the south (divided highway in a scenic valley) to mishmash of roads the cross the smaller river in the north. Another scenic stretch is near the Palisades state park in Illinois.
https://experiencemississippiriver.com/US 61 follows the river most of the ways, and in many parts is the 'official' river road (or one of several alternatives on one side or the other).