Is $\mathrm{End}_R(R[x])$ (the set of all $R$-module homomorphism from $R[x]$ to $R[x]$) isomorphic to $R[x]$, for some ring $R$?
Also, Is $\mathrm{Hom}(R[x],R)$ (the set of all $R$-module homomorphism from $R[x]$ to $R$) isomorphic to $R$, for some ring $R$?
(Below $R$ should be commutative or else it's quite unclear what is meant by $R[x]$.)
$R[x]$, as an $R$-module, is just the free $R$-module on countably many generators $\{ 1, x, x^2, ... \}$. That means $\text{End}_R(R[x])$ is the ring of infinite column-finite matrices over $R$, which is a bigger ring than $R[x]$, and $\text{Hom}(R[x], R)$ is $\prod_{i=1}^{\infty} R$, which is again bigger than $R$.
However, it's true that the set of all $R$-algebra homomorphisms from $R[x]$ to $R$ is just $R$. Abstractly this means that $R[x]$ is the free $R$-algebra on one generator. It's also true that the endomorphism ring of $R[x]$ as an $R[x]$-module is $R[x]$. In both cases you need to explicitly remember more structure involving $R[x]$ as a ring.