I can only imagine the mess if private companies disagree as much on how to build a road as railways did on gauges.
Would there be a standard side of the road to drive on, or would interchanges have to deal with switching from right-to-left side driving if a British designer/company owned a road? It might even lead to cars having steering wheels in the centre of the vehicle if that were the case.
I can see how roads were owned by the post office due to an unusual interpretation of the constitution when it mentions postal roads. Split the post office due to monopoly concerns later on and you have multiple Road companies. Also make every town, enclosed suburb/body corporate own it'd own roads and have farming cooperatives and large companies own the branch connections to the nearest post Road.
This all seems dystopic, but somehow no worse than electric power, water, sewage, railway, phone, etc networks being private. Sure eventually the government will impose standards and oversight to simplify the mess, but only once the whole underlying structure is set in stone.