I live in the Tallinn old town and just realized that many streets are named after famous buildings. For example, the street right next to me is named Sauna, because there used to be a sauna there in medieval times. Not sure what it was called before that or whether it had a name at all. So... How did street names become a thing?
In Japan, they don't name the streets but rather they number the city blocks, so addresses would be something like block 10, building 4.
Street names originated in whatever was the most important landmark located there. A document from 1166 in León, Spain, in which the abbot of San Isidoro exchanged lands owned by the monastery outside the walls of León for property next to their pilgrims' hospital established the limits of the area being received by the monastery in the following terms:
And we accept from you all that is below the walls and is next to the hospital of San Isidoro and is thus determined: from the first part situated by the aforesaid hospital; from the second of the city; from the third those towers; from the fourth the street that goes from the castle to San Isidoro. (Charter published in Encarnación Martín López, Patrimonio cultural de San Isidoro de León: Serie documental, documentos de los siglos X-XIII, colección diplomática. León, 1995, pp. 111-112.)
In this text, you can recognize the orality of the ways to which medieval streets were referred. Streets with a saint's name usually refer to a church or monastery once at that location, while a guild or vendor's name (Baker, Chandler, Wright) gives an indication of the goods that could be purchased on that street.