What does it mean for a sequence of random vectors to converge to a random vector? I am reading about convergence of random variables from Wikipedia and I come across this.

Note that the condition that $Y_n$ converges to a constant is important, if it were to converge to a random variable $Y$ then we wouldn't be able to conclude that $(X_n, Y_n)$ converges to $(X,Y)$.

This makes me conclude that
$\lim_{n \to \infty}(X_n, Y_n)$ and $(\lim_{n \to \infty}(X_n), \lim_{n \to \infty}(Y_n))$ are different things. Could you please help me understand why the two are different? Would really appreciate an example

The distribution of $(X,Y)$ is not particularly well defined here unless one is almost surely constant.  In general you would need to know more than $\lim\limits_{n \to \infty}X_n$ and $\lim\limits_{n \to \infty}Y_n$ can tell you.
For example suppose $X \sim N(0,1)$ and $Y \sim N(0,1)$.  You would not know from this whether $X$ and $Y$ are always equal to each other (correlation $1$) or independent of each other (correlation $0$) or something else.
So you would not know  the distribution of $(X,Y)$ or the limiting distribution of $(X_n,Y_n)$, at least without further information.
