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?

Suppose that $X\sim Unif(0,1)$ and $Y\sim Unif(0,X)$, then the joint distribution will be
$$f(X,Y)=f(Y|X)f(X)=\frac{1}{x}, \ \ \ 0\leq y \leq x\leq 1$$
If I'm not mistaken a uniform distribution gives the same importance to the whole domain of $(X,Y)$, whereas in the case displayed the importance given depends on the sampled $X$.
