Definitions, yes. Axioms not necessarily.
Note that when we are looking for a definition of $X$, we ask: "What are the necessary and sufficient conditions for something to be an $X$?"
If $P$ is a sufficient condition for $Q$, we write $P \rightarrow Q$
If $P$ is a necessary condition for $Q$, we write $Q \rightarrow P$
So, if $P$ is a necessary and sufficient condition for $Q$, we get $(P \rightarrow Q) \land (Q \rightarrow P)$, which is of course just $P \leftrightarrow Q$
So that's where the biconditional comes from in case of definitions.
Axioms can be used to capture definitions as well, in which case we call the definitional axioms. But not all axioms express definitions.