Is inclusion map not the same as identity? Wikipedia says $i$ is an inclusion means $i: A \to B$ with $A \subset B$ means $i(x) = x$ for each $x\in A$. 
But doesn't this mean $i(A) = A$, so this is actually identity?

No it isn't the identity since it isn't surjective. In the case when $B=A$ then this injection is the identity.
