This is China where an intercity highspeed train can be described as light rail that we are talking about, it isn't as confusing as that
As for why not do it as state level, I think the idea was kinda like compete against routes operated by state rail. Currently, all train tickets in China have to be brought ahead of time with specific train number and seat assigned, so you can't just go to a train station and hop onto any trains as you wish like any other metro/city train in other part of the world.
Also, the national train company/department's local subsidiary in that area have been empowered to determine train ticket pricing according to market demand, which translate to higher ticket price, therefore current higher speed train travelling between Guangzhou and Shenzhen which are about a hundred km apart cost ~11USD while high speed trains travelling between the two cities cost ~14USD, both of them are considered hella expensive by Chinese standard and is believed to be inhibitory to economic development in the area, therefore there are desire for cheaper and more seamless transit option.
Of course, there are also additional new rail route(s) in construction by the national train company that connect between these area, so it would probably be faster for cities to plan and build it themselves.