How rich and powerful were Italian medieval and Renaissance banking and merchant families? How do they compare to contemporaries in other parts of Western Europe? Did the Muslim world have equivalents in the middle East and North Africa?

Extremely rich and powerful, since they became the ruling families of more than a few Italian states. Certainly other banking families in the Lowlands and Germany became equally rich, but I don’t know of any that became as politically powerful other than the Fuggers. 

There certainly were wealthy merchants in the Muslim world, although banking was complicated by religious prohibitions. I don’t know of any who gained the same political power as the Medici did, however.