If a random generator function is not supplied to the random_shuffle algorithm in the standard library, will successive runs of the program produce the same random sequence
25.2.11 just says that the elements are shuffled with uniform distribution. It makes no guarantees as to which RNG is used behind the scenes (unless you pass one in) so you can't rely on any such behavior.
In order to guarantee the same shuffle outcome you'll need to provide your own RNG that provides those guarantees, but I suspect even then if you update your standard library the random_shuffle algorithm itself could change effects.