According to the Quran, Muslim men could have as many as four wives if they were rich enough to take care of them all. But most men in the Islamic world still had only one wife. In fact, most people in the Islamic empire lived the way people had in the Roman Empire and the Sassanian Empire before them. They lived in families with a dad and a mom and a bunch of kids. Sometimes the dad’s mother and his unmarried brothers and sisters (the kids’ aunts and uncles) lived with them.
In those days, the government didn’t let men marry men, or women to marry women, as they can today. But there were many non-traditional families anyway. There were households where the dad had left, or the mom had died, or where unmarried brothers or friends lived together. And not everyone lived in a family: Lalla Arifa, for example, left her family to live in a Sufi monastery for women. People could get divorced pretty easily, much easier than in Christian Europe.
As in earlier times, other people also lived in the house and counted as part of the family. These were mostly slaves or employees. People in the Islamic Empire, like the Sassanians and the Romans before them, still didn’t see much of a difference between their families and their businesses.