The existence of such a standard doesn't automatically guarantee compliance. There are plenty of APIs outside the scope of POSIX, plenty of places where POSIX has very under specified behavior, and even then, the compliance test suite doesn't test all of the rules and you still get tons of incompatibilities.
POSIX was, for the most part, not a major success. The sheer dominance of Linux monoculture makes that easy to forget, though.
Of course it doesn't guarantee compliance, but like all standards it makes interop possible in a predictable way, e.g. some tcsh scripts run fine under bash, but that's not by design. The inability or unwillingness of concerned parties to adopt the standard is a separate problem. This is why "posixly" is an adverb with meaning here.
POSIX was, for the most part, not a major success. The sheer dominance of Linux monoculture makes that easy to forget, though.