I know only about identities and quasi-identities as axioms of structures called "algebras". Are there other kinds of such formulas?

I perceive identities and quasi-identities as specific for algebra, because they define varieties and quasi-varieties, which can be defined also in terms of homomorphisms, subalgebras, products and subdirect products - notions specifically algebric. Are there other algebraic notions same strongly correlated with fundamental formulas describing them?

This question was motivated by my another question, where I named "adjunction algebra" an algebra with a binary operation and axioms among which one is not an identity, and I don't know whether or not it is equivalent with one or several quasi-identities.