POSIX does not specify a kernel interface, so Linux is largely irrelevant. It does specify the system interface, various tools, and extensions to the C standard, which could exist on top of any kernel. It is not POSIX-compliant in the sense that it's not mentioned, or it is POSIX-compliant in the sense that it's not mentioned, at your option.
There are UNIX®-Certified Linux distributions, so it is certainly possible to have fully POSIX-compliant operating systems using Linux. Huawei's EulerOS is one that has and that you can buy if you'd like. Most of the rest haven't paid their money and so don't have access to the test suite to check conformance.
Whether they would satisfy it in practice is not clear, but some do try harder than others. I suspect that some of the BSDs are closer than most Linux distributions, but that's a guess: for example, I know that `execlp("cd", "/", NULL)` fails on most Linux distributions, but works on many BSDs and is required by POSIX.