Are elements of a sample i.i.d. realizations of the same random variable or realizations of different i.i.d. random variables?

If there is some sample $X^n=(x_1,x_2,\dots,x_n)$, do we consider the elements of this sample $x_i$ independent and identically-distributed realizations of the same random variable $X$ or are they all realizations of different independent and identically distributed random variables $X_1,X_2,...,X_n$ (observation $x_1$ is the realization of a random variable $X_1$, observation $x_2$ is the realization of a random variable $X_2$ etc.)?

I hope this makes sense.

You may think about $(x_1,\ldots,x_n)$ as a realization of $n$ independent copies of $X$. Basically, there is a probability space $(\Omega,\mathcal{F},\mathsf{P})$ in the background so that $(x_1,\ldots,x_n)=(X_1(\omega)\ldots,X_n(\omega))$ for some $\omega\in\Omega$, which is chosen randomly according to $\mathsf{P}$. Then the statement "independent and identically-distributed realizations of the same random variable" doesn't make sense. Although, sometimes $(x_1,\ldots,x_n)$ is referred to as a random sample from a particular distribution (e.g. $F_X$).