If planes are around forever, wouldn't you be better off interning them? e.g. having a single global std::vector<Plane> (or std::array<Plane, N>) and passing around offsets in that array? And your PlaneVec would just be a glorified std::vector<size_t> (or int)? I don't see any value in maintaining a reference count if you're never intending to clean up these objects.
(The argument for using int here would be if you always have fewer than 2 billion planes, and so you can store a PlaneVec in less space. size_t is indistinguishable from Plane* in this context; you have the same amount of indirections either way.)
As I said, shared ownership has its uses, but most instances I've seen could have been replaced with a different model and would have been less painful to debug memory leaks and use-after-free.
