Why do we have to make variables unique when evaluating $\lambda$-calculus?
When performing a $\beta$-reduction, there are cases in which an $\alpha$-conversion is needed first; otherwise, the $\beta$-reduction fails to preserve semantics. This is here the case: you are replacing the free variable $x$ with $e_2$ and $e_2$ contains free variables which are bound within $e_1$.

You can easily see why the semantics are not preserved: $(\lambda y.y)$ and $(\lambda y.x)$ are not equivalent expressions because $(\lambda y.y)z = z$ but $(\lambda y.x)z = x$. In order to preserve the semantics, you first need to perform an $\alpha$-conversion on $(\lambda y.x)$ so as to replace $y$ with something else which is not free in $e_2$ (thus achieving "uniqueness"—though IMO such a term should be frowned upon here).