I saw someone on Reddit ask if Ur had street names. Yes, Ur had street names and it wasn't the only city to have them. Having street names was essential for economic and legal documentation, ritual instruction, and civic recognition.
In a text from Nippur dated to Damiq-ilishu (1816-1794 BCE), Inim-kugani's children were allotted numerous properties: house plots, storehouses, granaries. There were at least 7 properties listed, distinguished by either neighbors or street:, e.g. "next to Nur-Sin's house on Great Gate's boulevard."
We also know of some street names because they were important for festivals and processions. The Ay-ibūr-šabû in Babylon brought folks and gods alike directly into the downtown temple district through the Ishtar Gate. The road's name means "May the arrogant not flourish!"
Those roads were invested by the crown palace: "From Ištar-sākipat-tēbîšu to Kasikilla, the wide street Ištar-lamassi-ummānīša... from lkkibšu-nakar to Nabû’s entrance into Esagil, the wide street Nabû-dayyān-nišīšu... I made a massive infill and improved the road with bitumen and baked brick."
The inscription above was not discovered at Babylon, but Wadi Brisa (modern day Lebanon), 900 kilometers away. Why would folks there care about Babylon's roads? According to Nebuchadnezzar, in order to properly venerate his gods, everything had to be splendid from temple to road, brick by brick.
So, yes, Ur and just about every other city in the ancient Near East had street names. Moreover, they were maintained by the state and were points of pride for rulers and points of veneration for the gods. And not just roads had names but canals, gates, walls, buildings and fields, too!
More on street names: Buzû Street.