In practice, there is a stable and standard ABI in C++. On Windows systems, you use the (undocumented, but mostly reverse-engineered) MSVC ABI. On everybody else, you use the confusingly-named Itanium ABI, except for exception handling on ARM, which uses a slightly different ABI.

Standard library implementations do not necessarily share the same ABI, but it is possible to interoperate between libc++ and libstdc++, and libstdc++ maintains ABI compatibility across different versions.

And, to be pedantic, C does not provide a stable, standard ABI. It is the platforms themselves that provide this ABI standard, and these platforms equally provide a C++ ABI standard. So C++ ABI compatibility is no more or less a problem than C ABI compatibility.