The POSIX-conforming shell on AIX is the Korn shell and the Korn shell includes the keyword [[. So, I assume that you can use [[ in AIX's sh which is based on or linked to ksh.

The text from the POSIX shell command language you pointed to above says:
So, the POSIX standard allows [[ expression ]] to be supported by conforming shells, but says absolutely nothing about what [[ expression ]] does in a shell script.

The next revision of the POSIX standards and the Single UNIX Specification (which are likely to be balloted by IEEE, ISO/IEC, and The Open Group in about 3-7 years) is, in my personal opinion, likely to add defined behavior for the keywords [[ and ]] and many, or all, of common set of expressions that are available in both recent bash and ksh93 shells with [[ expression ]]. You can see a very lengthy record of a small subset of the discussion on this topic in The Austin Group's bug report on this topic: Bug #375: Extend test/[...] conditionals: ==, <, >, -nt, -ot, -ef. Note that this bug report is still open AND that the initially suggested Desired Action will NOT make it into the standard.