All souls are eternal. They don't die. There is no death as we know it on the soul level of reality. When a human dies, the lights don't go out. That is based on the Newtonian and Cartesian philosophies that models man after a machine that many here still believe is true. These ideas are outmoded and incorrect. The light actually comes on when a human dies.
Why do you think there is a belief in " old souls" here? How can you be an old soul if you have not grown soulfully? As a reminder, "the old soul is sewing the seeds of light on the carpet of linear time, not even knowing he will also be harvesting that exact mature plant of wisdom as he returns in a subsequent life."
An old soul is not the old as we know it in human terms. An old soul on the spiritual level is a soul that has lived many, many lifetimes and through life experience, has garnered much awareness. Old equates to expanded awareness in the soul arena.
Let me give you an example of expanding your soul's awareness. Let's say that you become an actress in a lifetime and your focus is all on yourself as in for myself alone as it must be to achieve success. You have to constantly work on your physical instrument, always look good, take classes, etc. But then something happens and it shifts your trajectory to become a High School theater teacher where you positively affect the lives of many students to become more enhanced human beings. In this case, your soul has gone from me, myself alone to service to other-self. This brings the soul back into balance and gives the soul an expanded awareness of being in service to others. The soul's awareness is expanded.
This we do on and on. You are a soul having a human experience, a soul which is eternal. So, you are many humans at once with a soul that contains the awareness garnered from the experiences of all those life times. This lifetime is just a point of light that has behind it all the other lives you have lived.
You are really one life. The eternal life of your soul that continues, undaunted through the annals of the now.
You are a continuum, not a limited 3 D speck of dust. You are eternal, now start acting like it.