Let $c$ be a characteristic class for principal $G$-bundles and $p_1: E_1 \to X, p_2: E_2 \to X$ be isomorphic principal $G$-bundles, then $c(E_1) = c(E_2)$
Is this part of the defining naturality condition for characteristic class i.e. part of the definition or it can be proved from the definition of characteristic class?
Well, this depends what definition you give. It is certainly reasonable to include this in the definition explicitly, and some definitions do (such as Wikipedia's definition). But depending on how you define "pullback", it could follow from the pullback invariance condition. Specifically, you might formulate pullback invariance as follows: whenever $f:X\to Y$ is a map and $E$ is a bundle on $Y$ and $f^*E$ is any pullback of $E$ (i.e., any bundle over $X$ that satisfies the categorical definition of the pullback), then $c(f^*(E))=f^*(c(E))$. Then if $E_1$ and $E_2$ are isomorphic bundles over $X$, an isomorphism $E_1\to E_2$ realizes $E_1$ as a pullback of $E_2$ with respect to the identity map $Id:X\to X$. Thus pullback invariance would imply $c(E_1)=c(Id^*(E_2))=Id^*(c(E_2))=c(E_2)$.