Let $U$ be the discrete uniform on $\{0,1,\dots,n\}$. If a discrete random variable $X$ satisfies:
$$ \begin{align} \mathbb{E}[X] &= \mathbb{E}[U] = n/2 \\ \mathrm{Var}(X) &= \mathrm{Var}(U) = ((n+1)^2-1)/12, \end{align} $$
is this sufficient to conclude that $X \equiv U$? For the case of $n=1$ and $n=2$ it can be shown to be true.
Addition: $X$ is known to have full support over $\{0,1,\dots,n\}$.
No. For $n=3$: consider the distribution of $X$ given by $p=(1/5,2/5,1/10,3/10)$. (There are many other possible choices.)
You have $$ \mathbb{E}[X] = \sum_{k=0}^3 k p_k = 0\cdot \frac{1}{5} + 1\cdot\frac{2}{5}+2\cdot\frac{1}{10}+3\cdot\frac{3}{10} = \frac{3}{2} = \mathbb{E}[U_3] $$ and $$ \mathbb{E}[X^2] = \sum_{k=0}^3 k^2 p_k = 0\cdot \frac{1}{5} + 1\cdot\frac{2}{5}+4\cdot\frac{1}{10}+9\cdot\frac{3}{10} = \frac{7}{2} = \mathbb{E}[U_3^2] $$