A language is the source code you write when building an application, while a library is a packaged set of features you can use in that language.
PHP has its own syntax, keywords, methods, etc. that you can organize into features, and organize those features into an application.
It's important to separate the compiler/interpreter process from the code you write. How the code is compiled or interpreted is a hidden implementation detail. You shouldn't have to worry about it.
Why is PHP not a library of C? Again, knowing that the PHP interpreter is written in C doesn't matter. You could write your own interpreter in C++ or Go but it wouldn't change the PHP language.
As another example, the gcc compiler is written in C, but that doesn't make C a library of itself. The same goes for C#, in which the Roslyn compiler is written in C#, but C# remains a language.