I'm only thinking in terms of Linux distributions since I never needed to deploy software on other UNIXes (excluding macOS, but Apple constantly forces changes anyway).

Do other UNIXes have any problems similar to glibc ABI problems that Linux users experience, or do they stabilise the libc ABI similar to how Linux keeps syscalls stable?

There are naturally ABI breaks between major OS versions, outside of what POSIX requires.

I will be even more impressed with linux syscall stability if your implication is that (some) people need to recompile their software for each major update on all other UNIXes.

Linux is only friendly for FOSS projects, it hardly has the same stability as commercial UNIX systems for closed source software.