Let $X$ and $Y$ random variables, $\{X_{i}\}_{i=1}^{n}$, and $\{Y_{i}\}_{i=1}^{n}$ samples. Is $\{(X_{1},Y_{1})\}_{i=1}^{n}$ a sample of $Z=(X,Y)$?

Question

Let $X$ and $Y$ two random variables with density distribution function $f_{X}$ and $f_{Y}$ respectively. We assume that $(X,Y)$ is a random variable on $\mathbb{R}^{2}$ where $$\mathbb{P}((X,Y)\in A)=\iint_{A}f_{X}(x)f_{Y}(y)dxdy. \tag{*}$$

Let $X_{1},\ldots,X_{n}$, and $Y_{1},\ldots,Y_{n} $ samples of $X$ and $Y$ respectivelly.

The question: Is $(X_{1},Y_{1}),\ldots,(X_{n},Y_{n})$ a sample of random variable $Z=(X,Y)$?

If the answer is negative, how from $X_{1},\ldots,X_{n}$, and $Y_{1},\ldots,Y_{n} $ I generate a sample of the random variable $Z=(X,Y)$?