They're trying various options to share some limited data (like address data) [1], but haven't gotten very far. They've done some work on "Community TIGER" [2], which aims to give validation information back to local governments for geographic data, but not the improvements that Census generates as part of the decennial census.
The US Postal Service has no motive to share with the Census Bureau. For one thing, the USPS makes all of its money selling limited access to its address list to advertisers. Additionally, the USPS's address list (or delivery points) doesn't necessarily correspond with people and where they live.
PS: I'm interested in this sort of thing because I help run OpenAddresses [3], a community-built list of authoritative address data sources from around the world. There's a lot of data out there!
[0] https://www.law.cornell.edu/uscode/text/13/9 [1] https://fcw.com/articles/2011/09/14/census-bureau-title-13.a... [2] https://www2.census.gov/geo/pdfs/gssi/Community_TIGER.pdf [3] https://openaddresses.io/
Openaddresses.io is an awesome project, as a GIS professional thank you! This type of data can be immensely useful!
http://www.columbia.edu/~fdc/postal/ is a pretty good overview of most of the postal addressing systems around the world.
Then you have situations where a street name is the same as the region/state (is the source data just missing attributes?), confusion about the direction prefix/suffix (1 W Hacker Street, 1 Hacker Street W - can vary within the same city on the same street), poorly named streets like "1 W Hacker Street East or "1 Hacker Street East W", or street name types that are not commonly used (1 Hacker Launch Pad etc).
It's even more fun when you start talking about countries that either use older addressing formats (I've seen addresses in Ireland that are just like (Old Blue Cottage, Some Town) or are not in english.
There are many more situations, these are just what immediately come to mind.