Why is the identity element so important in this construction. I looked up some books and notes but still do not see why. How could the construction started from tangent space of a element other than identity possibly fail to get a Lie algebra so that people can only get it from the tangent space of identity?
If $G$ is a Lie group and $g \in G$ then the map $L_g\colon G \to G$ defined by $x \mapsto gx$ (so, left multiplication by $g$) is an isomorphism (a topological isomorphism, it's not a group homomorphism). It's derivative gives an isomorphism between the tangent space $T_1G$ of $G$ at the identity and the tangent space $T_gG$ of $G$ at $g$.
So to answer your question, it's not special. The tangent spaces at all points of $G$ are isomorphic. So we just pick one to work with and the identity is the only element that every group is guaranteed to have, so we pick the identity.