There was a story on Russian news a few weeks ago how there are multiple streets with the same name in the city of Sochi. They are tackling the problem before the olympics arrive, but your assumption is probably violated elsewhere as well.
London post codes are the result of London moving from a single post town to a town with multiple towns (for the purpose of postal sorting offices).
The whole concept of a central post office broke down in the 1850s and London needed to be sub-divided to solve that.
Postal codes merely solved the routing problem between divisions, and wasn't a response to the street names problem.
One of the little known "Did you know" is that the primary numbers of postal areas in London were originally alphabetically sorted (which largely remains the case today unless that postal area has been extended or further sub-divided).