Are there a set of skill levels for playing piano and what is needed to master these levels?

Any music teaching organization is likely to develop structures of the kind you suggest. The problem in finding a single satisfactory answer to your question is that all these structures are to some extent arbitrary. Teachers, and organizations of teachers, may hold the same view on the general sequence of topics but disagree on the details. However, all roads are likely to lead to Rome.

If you're looking for examples of structured skills and repertoire, you might look at a syllabus of examinations. Some organizations dispense external examinations, that is outside their own rooms by itinerant examiners. For example, here in the UK we have bodies like the ABRSM (Associated Board of the Royal Schools of Music), and Trinity College London, conducting graded examinations in many countries in voice and various instruments.

Google something like "abrsm piano grade exams syllabus" and browse the requirements. Then look at a syllabus from another provider and compare. Probably, you will see a broad similarity in skill progression, but with different arbitrary choices of names for the levels of skill. "This is a grade four piece!" - "No, it's grade six!" - "Oh, you're with THOSE people." (It's the same piece...)

Obviously, you needn't be taking examinations to benefit from the sequence that the syllabus suggests. Some people find these structures useful for organizing their work and getting a feeling of progress. All the same, while progress in an instrument often feels like it features plateaus, and sudden leaps forward, it is essentially continuous not discrete.