Well, although that's a nice metaphor for the answer, in practice I can imagine that using a personally unique ID for physical mail routing is quite possible. A billion or so rows in a database that links an ID number to spatial coordinates is perfectly conceivable, it'd only be in the tens of gigabytes. It'd also aid in redeliveries and redirections. There would be no reason to even keep this unique, one could have many such numbers to avoid correlations.

Using the SSN itself, of course, has vast privacy & security complications.

Why doesn't this work for IP? Because on that scale, the route table is too large to advertise and too large to keep in the working memory of current router linecard ASICs for sub-microsecond lookup, speeds a postal service does not aspire to.