It's worth noting that postal unions on Earth historically tended to not become governments - Alliances, but not governments.
At present, one dominates on Earth. The UPU has been around since the 1874, and, except during a few wars, has managed across-border handoffs of mail between even hostile members.
Note that, post WWII, the IPU was placed under UN authority.
There are even a couple gags about the mail getting through during the shadow war in Babylon 5...
That a Postal Union could become a government of their own? Depends upon how large the postal union is, how small the members are, and if they can be coaxed into funding a serious level of forces to defend the mail.
Postal Unions start to look particularly government-like when they are able to establish a transport monopoly... The Spacing Guild in Dune is almost a shadow government. Note that Edric can, essentially, coerce even the Padishaw Emperor. (Note also: Herbert later realized his logic errors - the spacing guild should have become the empire, given the setup, by simply dropping Arrakis off the map entirely until everyone surrenders. So, he introduced ships that didn't need Spice nor Guild Navigators, and had them banned by convention... which holds the Guild's monopolies in check.)
Key assertion: Any hydraulic despot who controls also the interstellar communications and interstellar transport controls the government.
Corollary: He who controls the postal union in the absence of FTL Comm has a strong leg on the government already.
Corollary and secondary assertion: He who controls FTL comm controls the government, as long as it's faster than shipping.
FTL comm monopoly leading to quasi-government is a common enough trope. Half a dozen short stories and novellas, plus a few games...
Edit: One other thought - the classic "Postal Union as Empire" trope usually leads to confederation type feel - the Postal Union makes certain demands, and aside from them, ignores the locals. Which said demands could include creeping increases in authority over time.