How come identity encodes absurdity
looks entirely identity function to me, which can definitely be inhabited by a closed term


A proof of $\forall C : Prop, C$ is a function from an arbitrary proposition to its proof. Since a proposition isn't a proof, an identity function is not a function from propositions to proofs.

The identity function (or rather, a function returning the identity function on proofs of $C$) corresponds instead to the proof of $\forall C : Prop, C \to C$. That's even the example on the beginning of page 3!