As you say, a sentence of the propositional logic is valid if and only if it is satisfied by every truth assignment. So the sentence "Φ ↔︎ Ψ" might be valid if the sentences Φ and Ψ are such that there is no valuation under which one of them is true and the other false. For example, if Φ is ¬(A ⋀ B), and Ψ is (¬A ⋁ ¬B) then Φ ↔︎ Ψ is valid, since there is no way to assign truth values to A and B that makes Φ ↔︎ Ψ come out false. Other examples of Φ ↔︎ Ψ might be invalid.
A more general answer to your question would be to point out that each logic comes with its own specification of what constitutes validity. Validity in propositional logic is simply a matter of assigning truth values to the propositions. In predicate logic things become more complex and we speak of propositions having interpretations under which they may be true. Then there are different logics entirely, such as intuitionistic logic, which have different rules for validity. There is also an important distinction between syntactical and semantic validity. In simple terms, syntactical validity is concerned with the rules that determine whether something is provable just by manipulating formulas, while semantic validity is concerned with whether a semantic property (usually truth) holds under all interpretations, or all possible worlds, or some other generality.
Edit: it is worth clarifying that it is obvious from your question that you are asking about the concept of validity as it occurs in the context of logic and reasoning. In ordinary English 'valid' has other uses, as one can speak of a valid will, or a valid ticket for a journey, or a valid contract. Statisticians also describe a data set as valid, meaning that it is unbiased and correctly represents what they are trying to measure. Those are quite different uses of 'valid' from the way logicians use the term.