First of all, please note that there was no Muslim society until the early 600s AD. It was only then that Islam even came to exist and it did not come to be the dominant religion in any large area until around 630. In addition, please note that any discussion of such a large society for such a long time must be very generalized.
In general, the biggest distinction in the Muslim world was between slave and free person. Slavery was very widespread in the Muslim world. Muslims could not keep other Muslims as slaves so slaves were people from other religions and cultures, many of them captured in wars.
A somewhat related distinction was between believer and non-believer. Islam conquered such a large area that many Muslim societies had non-Muslims within them. These non-Muslims were generally treated well but had to do things like paying for the privilege of keeping their own religion.
Finally, as with most large societies, there were class differences. There was an upper class that consisted largely of the rulers of societies. These were the "royalty" as well as the high government officials. Some of the wealthiest merchants were part of this class as well.
So, Islam during this period has social differences based on religion, on wealth, and on slavery.