Does the International law forbid countries to recognize the
sovereignty of any political formation that is not recognized by all
the members of UN?
And vice versa: Does the International law obliges countries to
recognize the sovereignty of any political formation that is
recognized by all the members of UN?
No and no. There are six U.N. member states that are not recognized by at least one other U.N. member state. And there are at least six states that are not U.N. members that are recognized by at least one U.N. member state.
Whatever the answer, can you, please, provide a link to the
corresponding clause in the International law?
There is no such thing. International law is a set of general principles, often internally inconsistent, that guide the diplomatic and military affairs of sovereign states, as I explain in the first part of my answer here. It isn't "law" in the sense that ordinary domestic law is "law".
There are treaties and treatises out there that inform efforts to articulate what constitutes international law, but there is no world legislature out there, and there is no court with the power to issue decisions to any country in the world that will be observed without the use of military force on a wide array of issues.
Countries judge their own cases in international law, and do so on a case by case basis, not based upon grand universal principles that clearly dictate an answer. Your question has an answer that is best demonstrated by example, rather than by reference to any particular legal authority. And, in general, international law's force and sources have more to do with examples and analogies than with an authoritative statute-like rule book.