What is a sample path of a stochastic process

The image shows five sample paths of a geometric Brownian motion process $\{X_t\}_{t\ge0}$. The paths are different functions of $t$ (which you may think of as representing time). Each one of them shows the values that $X$ takes over time under a specific outcome ($\omega_1,\dots,\omega_5$) in the sample space $\Omega.$

Notice by contrast that the expected value $E[X_t]$ of the process is also drawn (as a smooth function of $t$) and that it is not a sample path; rather than being the result of any single outcome, it is obtained by "averaging" over all of them according to the law of the process.

Each random variable $X_t$ is an $\mathcal F$-measurable function $\omega\mapsto X_t(\omega)$ from $\Omega$ to $S$. The stochastic process $X$ may therefore be viewed as a function of two variables $\Omega\times T\ni (\omega,t)\mapsto X_t(\omega)\in S$. For each fixed $\omega\in\Omega$ we have the partial function $t\mapsto X_t(\omega)$; this is the sample path of $X$ corresponding to the sample point $\omega$.

The problem is that it's simply not true that "then the value of each $X_j$ should be the same".

Hmm. Example: Say $\Omega=[0,1]^2$ with $P$ equal to Lebesgue measure. Define $X,Y:\Omega\to\Bbb R$ by $X(s,t)=t$ and $Y(s,t)=s$. Then $X$ and $Y$ are iid, but $X(\omega)\ne Y(\omega)$.