When modelling a sample $(x_1,\ldots,x_n)$ as an $i.i.d$ sample from a given distribution $F$, the correct way of modelling is to see this sample as the realisation of n random variables $(X_1,\ldots,X_n)$ made of $n$ independent random variables identically distributed from $F$:
$$(x_1,\ldots,x_n)=(X_1,\ldots,X_n)(\omega)\qquad\omega\in\Omega$$
The concept of $n$ realizations of a single random variable is a shortcut that is not well-defined because one cannot handle independence with a single random variable.
