Is there any difference between $\sigma((X_1,X_2))$ versus $\sigma(X_1)\cup \sigma(X_2)$? For random variables $X_1,X_2$, is there any difference between $\sigma((X_1,X_2))$ versus $\sigma(X_1)\cup \sigma(X_2)$?
If not, how to explain that $(X_1,X_2)$ is $\mathcal{F}$-measurable iff $X_1,X_2$ are $\mathcal{F}$-measurable?

Yes. For example, one of them is always a $\sigma$-algebra whilst the other isn't in general. Instead, we have $\sigma(X_1,X_2) = \sigma(\sigma(X_1)\cup \sigma(X_2))$.
