according to a couple of chess history books I have the consensus is that it was invented in india, then became popular in Persia. It changed a little to reflect the society that played it. The European version became the standardized version. That's it in a nutshell but I recommend reading a couple of them. "The immortal game" is very interesting and an easy read. It may have some embellishments. I'm reading another one that is much more thorough and a bit of a challenge to read. It leaves out nothing.
Anyone who knows