Most contemporary C++ compilers don't support anything other the extern "C" and extern "C++" required by the ISO C++ standard. Integration with other languages most commonly uses C calling conventions and extern "C".
And even when non-C calling conventions were supported, they often weren't selected by an extern string literal, they still used extern "C" but in conjunction with some non-standard calling convention declaration: for example, Microsoft C++ for 16-bit platforms (DOS, Windows 3.x, OS/2 1.x) used __pascal for the Pascal calling convention and __fortran for Fortran, instead of something like extern "Pascal" or extern "Fortran".
That said, IBM's contemporary C++ compilers for z/OS support extern "FORTRAN" for calling Fortran, extern "COBOL" for calling COBOL, and extern "PLI" for calling PL/I.
Similarly, IBM's ILE C++ compiler on IBM i supports a whole list of different extern string literals, see PDF page 338 of IBM i: ILE C/C++ Programmer's Guide, like extern "RPG", extern "C nowiden" (which is like extern "C", but turns off integer promotion, so a short will actually be passed as a 2 bytes instead of 4), extern "OS" (which enables IBM's legacy pre-ILE calling convention).
It appears, nowadays, IBM is the only major C++ vendor who supports non-"C"/"C++" extern, and even for them only on their mainframe and midrange platforms; but, historically, there probably were others.