As I see it, not in name but in reality it more resembles two departments of the same company working on a common project.
Don't be tricked by the seemingly two entities. The two are separate only in name but the same one in reality. It is not like Boeing vs. LM, but rather CAC vs. SAC under the same owner. The teams within them can be just moved to one another, same as the project ownership. Basically when observing Chinese space/military/SOE, one should drop the idea of independent competing commercial companies or factions although they do exist, but can be easily broken at will by CPC who is the only boss. We have seen that in China's telecom industry and railway industry repeatedly being broken apart and merged when the true owner sees the need.
Back to the specific program. As far as I know there is only ONE project that is the Combined Cycle Engine based carrier aircraft + space plane orbiter. It seems that CASC leads the program, but you can see that in its slide, the first step is CZ-8R which is a work of CASIC's 8th institute. This is one of the examples that they work together as one person.