[libc][NFC] Refactor internal errno.

This is in preparation for the transition to a solution to make libc tests
hermetic with respect to their use of errno. The implementation of strdup
has been switched over to libc_errno as an example of what the code looks
like in the new way.

See #61037 for more information.

Reviewed By: lntue

Differential Revision: https://reviews.llvm.org/D144928