Why does libc_nonshared.a exist? What purpose does it serve? I haven't been able to find a good answer for its existence online.
As far as I can tell it provides certain symbols (stat, lstat, fstat, atexit, etc.). If someone uses one of these functions in their code, it will get linked into the final executable from this archive. These functions are part of the POSIX standard and are pretty common so I don't see why they wouldn't just be put in the shared or static libc.so.6 or libc.a, respectively.
It was a legacy mistake in glibc's implementing extensibility for the definition of struct stat before better mechanisms (symbol redirection or versioning) were thought of. The definitions of the stat-family functions in libc_nonshared.a cause the version of the structure to bind at link-time, and the definitions there call the __xstat-family functions in the real shared libc, which take an extra argument indicating the desired structure version. This implementation is non-conforming to the standard since each shared library ends up gettings its own copy of the stat-family functions with their own addresses, breaking the requirement that pointers to the same function evaluate equal.