Platform specific POSIX C library

I understand that POSIX C library is used to make code, using system functionalities (system calls), portable among different OS adherent to POSIX specification.

But in more practical terms can we say that POSIX library functions are just a wrapper towards platforms specific System API ?

Is my understanding correct ?

If it so there should exsist many versions of POSIX C library,one for each platforms.
So I wonder why I didn't have to download a platform specific Posix library working in Ubuntu ?

thanks,and sorry if this questions is too trivial for you.