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What is the meaning of University, and why don't these old "universities" meet the criteria?
Religion, as an element of society that is discrete from our understandings of just how stuff in the world works, is a conception that was pretty unique to Christian civilization (and specifically Protestantism at that) which became generalized due to Western European cultural and economic dominance.
Most other times and places just didn't view them as being all that distinct. Their religious traditions didn't insist on strict separations between worldly pursuits and spiritual ones and often didn't really separate the formal institutions from each other. Most people would just adopt bits and pieces of all the various intellectual traditions from the priests and monks, but they'd adapt them into their personal or family practices. Normal people today would not try and identify themselves as "deontologists" or "utilitarians" today. They pick these ideas up from the nerds and applied them to their lives, but it wasn't really important for them to identify as one or another the way we insist folks split hairs as to whether they're Christian or Buddhist or Taoist or whatever now. These were intense (and sometimes violent) arguments among the intelligentsia, but not so much outside of that.