Suppose a sequence of random variables $\{X_n\}$ converges in distribution to some probability measure $\mu_X$ on $\mathbb{R}$, and similarly $Y_n \stackrel{d}{\Rightarrow} \mu_Y$.
When is it true that $(X_n,Y_n)$ also converges in distribution? Or, more generally, when is $(X_n,Y_n)$ tight?
On
If $(\mu_n)_{n \in \mathbb{N}}$ is a sequence of Borel probability measures on $X \times X$ endowed with the product topology for a Polish space $X$, then compactness of the corresponding marginal sequences $(\mu_n^{(1)})_n$ and $(\mu_n^{(2)})_n$ implies tightness of $(\mu_n)_{n\geq 1}$ as follows:
Let $\epsilon > 0$ and $K_i \subseteq X$ compact such that $\mu_n^{(i)}(K_i) \geq 1-\frac{\epsilon}{2}$ (such sets exist by the assumed tightness of both marginal sequences). Then, $K_1 \times K_2 \subseteq X\times X$ is compact and for each $n \geq 1$: $$\mu_n((K_1\times K_2)^c) \leq \mu_n(X\times K_2^c)+\mu_n(K_1^c \times X)=\mu_n^{(2)}(K_2^c)+\mu_n^{(1)}(K_1^c) \leq \epsilon.$$Hence, $(\mu_n)_n$ is tight.
This applies to your question by considering $\mu_n^{(1)} \sim X_n$, $\mu_n^{(2)} \sim Y_n$ and $\mu_n \sim (X_n,Y_n)$, noting that by your assumption both these marginal sequences are tight.
Tightness of marginals implies joint tightness in the product topology. This is readily seen from the definition of tightness, recalling that the product of two compacts is compact.