It's not true in general.

For example in the theory of lattices, some are distributive and some are not. So $$x\wedge (y\vee z)=x\wedge y\vee x\wedge z$$ is an identity that holds in some but not all lattices.

It seems the name for this idea is equationally complete theory, see page 30 of Walter Taylor's Equational Logic survey.

Not every theory is like that: for example in the theory of lattices, which is defined by axioms such as $$x\wedge (y\wedge z)=(x\wedge y)\wedge z$$ (the full list is at Wikipedia) some are distributive and some are not. So $$x\wedge (y\vee z)=(x\wedge y)\vee x(\wedge z)$$ is an identity that holds in some but not all lattices.