No distinction of individuated souls, if that's what the idea is dependent upon. Too many different people claiming to have once been the same dead celebrities or historical figures.
What we share is mental experiences and human traits which keep being revived and staying in circulation as long as we keep reproducing. Call it a more fundamental identity of generic subjectivity or just conscious experience itself, which underlies and binds us as a persistently reemerging marvel.
No personal memories are passed on either genetically or mysteriously to anyone, but the culture itself can teach existing general information to children. Scattered newborns gradually grow up into a spectrum of personality types that have existed many times before. Each experiences similar introspective and extrospective manifestations (constructs of qualia), and entertain classic thoughts as well arguably new ones.
The universe is filled with non-conscious matter which thereby is normally not even nothing to itself (invisible); and indeed even neural correlates of consciousness in a brain don't acquire any new, radical properties detectable in them (just matter interactions as usual). Only personal reports associate them with experiences.
So all sufficiently conscious organisms are, in a sense, kind of like islands of manifestations in a sea of absence. Each newborn "light" figuratively rises from the emptiness and maintains the continuity of phenomenal events in a general context. Though specific identities eventually perish, the new ones continue to rise from the blankness. Starting out with a common, generic subjectivity that incrementally re-acquires everything lost by death, but not assuming a particular or quasi-unique subject that existed once before in the past (no connective mechanism there).
