Bivariate Distribution with Uniform Marginals is Bound to be Uniform? If $X\sim U , Y\sim U$ , and $X,Y$ may be non-independent. Can we say the joint distribution of $X,Y$ is uniform?

No, the joint distribution is not necessarily uniform.
Consider $X$ and $Y$ with a joint pdf
$$
f(x,y) = \begin{cases}
2, \text{if } x \in (0,0.5), y \in (0,0.5)\\
2, \text{if }x \in (0.5,1), y \in (0.5,1) \\
0, \text{otherwise}
\end{cases}
$$
Then both $X$ and $Y$ have marginal $U(0,1)$ distributions, but the joint distribution is not uniform.
