None of the other answers (yet) mentions an important feature that promotes interworking between compilers and libraries - the ABI or Application Binary Interface. On Unix-like machines, there is a well documented ABI, and the C compilers on the system all follow the ABI. This allows a great deal of mix'n'match. Normally, you use the system-provided C library, but you can use a replacement version provided with a compiler, or created separately. And normally, you can use a library compiled by one compiler with programs compiled by other compilers.
Sometimes, one compiler uses a runtime support library for some operations - perhaps 64-bit arithmetic routines on a 32-bit machine. If you use a library built with this compiler as part of a program built with another compiler, you may need to link this library. However, I've not seen that as a problem for a long time - with pure C.
Linking C++ is a different matter. There isn't the same degree of interworking between different C++ compilers - they disagree on details of class layout (vtables, etc) and on how exception handling is done, and so on. You have to work harder to create libraries built with one C++ compiler that can be used by others.