Is the limit of a sequence of random variables unique?

The limits are unique - more precisely, they are unique in distribution, i.e. if $X_n \to Y$ in distribution and $X_n \to Z$ in distribution, then $Y$ and $Z$ have the same distribution. If you define a metric $d$ such that

$$d(X_n,Y) \to 0$$

if and only if $X_n \to Y$ in distribution, then you actually define a metric on the space of distributions. This means in particular that

$$d(X,Y) = d(\tilde{X},\tilde{Y})$$

whenever $X \sim \tilde{X}$ and $Y \sim \tilde{Y}$. From this it follows obviously that $X_n \to Y$ in distribution implies $X_n \to Z$ in distribution for any $Z$ which has the same distribution as $Y$.

In contrast, if $X_n \to Y$ in probability (or almost surely), then $Y$ is unique in the sense that whenever $X_n \to Z$ in probability, then $Y=Z$ almost surely.