No, reincarnation follows that the same soul is passed on to another life. Unification of souls is apparent in literally no spiritual beliefs.
In fact it's somewhat offensive that you hang out beliefs you can't even name (because you obviously haven't done your research, par the usual) for the sake of saving your stupid argument.
The closest thing to unification of "souls" in spiritual beliefs are that the Boddhisattva becoming one and the same upon reaching Nirvana, but there is no reference to "the soul" in that regard.