There are multiple branches within the New Age movement that assume that time is not real. This is an extension of Idealism thinking, where the presumption is that the mental world is what is real, and the physical is a somewhat delusion-based shared illusion. And if matter is an illusion, so is space-time.
What these idealist new age thinkers presume is that there IS a logical time, as in a real series of state sequences. But that this logic sequence is not coupled to the time dimension in our material world. This is therefore a time-prime, which would be real, even though the physical worlds time is not real.
Once you have this framework: idealism, illusory time, and a higher level of time-prime, then the possibility of all souls being identical, just at different stages of maturity (per time prime) become a possible hypothesis.
I have read a variety of new age thinking and have encountered this concept there. I cannot recall the specific references where I would have seen this.
I checked a couple of major New Age references: Eckhard Tolle, Seth/Jane Roberts, the Ra material, and A Course in Miracles, and all hold by idealism and illusory time, but none of those argued for a single soul traversing all of us in sequence. You will have to dig deeper to find the advocates of this speculation.
Aside: There are a few theoretical physicists who have applied a similar principle to electrons. We cannot distinguish one electron from another, and if "the identity of indistinguishables" is correct, then they are plausibly all the SAME electron at different points in its history. Physics equations also all work just fine both backwards and forwards in time, and an electron traveling backwards in time looks like a positron to us. Under Block Time, the present is an illusion, with no particular relevance, so one electron could travel back and forth thru all of time, and appear repeatedly in our apparent present as quadrillions of indistinguishable electrons and postitrons. Here is a summary: https://en.wikipedia.org/wiki/One-electron_universe