No. Consider the two different joint distributions on $X$, $Y$, both with values in ${0,1}$:
$$ P_1(0,0) = \tfrac12, P_1(0,1)=0, P_1(1, 0)=0, P_1(1, 1)=\tfrac 12$$
and
$$ P_2(0,0) = P_2(0, 1) = P_2(1, 0) = P_2(1, 1)=\tfrac 14$$
The two different joint distributions have identical marginal distributions (namely, both $X$ and $Y$ are uniformly distributed on $\{0,1\}$).
In your Gaussian example, $X$ and $Y$ could either be independently distributed Gaussians, or they could be the same variable -- or anything in between.
For any $x, y$ in $[0, 1]$,
\begin{align}
\Pr(X \le x, Y \le y)
& = \Pr(\sqrt U \le x, UV \le y) \\
& = \Pr\left(\sqrt U \le x, U \le \frac yV\right) \\
& = \int_0^1 \Pr\left(U \le x^2, U \le \frac yV \mid V = v\right) dv \\
& = \int_0^1 \min\left\{x^2, \frac yv\right\} dv \\
& =
\begin{cases}
x^2 & ; x^2 \le y \\
\int_0^{y/x^2} x^2 dv + \int_{y/x^2}^1 \frac yv dv & ; x^2 \ge y \\
\end{cases}\\
& =
\begin{cases}
x^2 & ; x^2 \le y \\
y - y \log y + 2y\log x & ; x^2 \ge y \\
\end{cases}
\end{align}
To obtain the probability density function, you take the derivative with respect to $x$ and $y$:
$$
f_{X,Y}(x,y)
= \begin{cases}
0 & ; x^2 \le y \\
2/x & ; x^2 \ge y \\
\end{cases}
$$
Best Answer
I will try to address the question you posed in the comments, namely:
Gives $0<x<2$ and $0<y<2$, we first compute $F_{X,Y}(x,y)$: $$\begin{eqnarray} F_{X,Y}(x,y) &=& \mathbb{P}\left(X \leqslant x, Y \leqslant y\right) = \mathbb{P}\left(U+V \leqslant x, U+W \leqslant y\right) = \mathbb{E}\left(\mathbb{P}\left(U+V \leqslant x, U+W \leqslant y|U\right)\right)\\ &=& \mathbb{E}\left(F_V(x-U)F_W(y-U)\right) = \mathbb{E}\left(\min(1, \max(x-U,0))\min(1, \max(y-U,0))\right) \end{eqnarray} $$ The pdf, being a derivative of the cdf, will then read: $$ f_{X,Y}(x,y) = \frac{\partial}{\partial x} \frac{\partial}{\partial y} F_{X,Y}(x,y) = \mathbb{E}\left( [1+U>x>U] [1+U > y>U] \right) = \mathbb{P}\left(U < x < 1+U \land U < y<1+U\right) $$ The evaluation of the latter expectation is straightforward but tedious, so I asked Mathematica for help: