Let $X_1,X_2$ be random vectors such that $F_{X_1}=F_{X_2}$. ($F$ denotes CDFs)
Let $Y_1,Y_2$ be random vectors such thar $F_{Y_1}=F_{Y_2}$.
If $X_1,Y_1$ are independent, then are $X_2,Y_2$ necessarily independent? I guess this is obviously false, but since I am new to probability theory, I am not sure how to construct such one. What would be a counterexample?
Distributions are not dependent or independent; random variables are. Dependence or independence is determined by the joint distribution, which his determined by the joint c.d.f. $F_{X_1,Y_1}.$
Consider these probability mass functions \begin{align} & \begin{array}{ll} f_{X_1,Y_1}(0,0)= 1/4, & f_{X_1,Y_1}(0,1) = 1/4 \\ f_{X_1,Y_1}(1,0) = 1/4, & f_{X_1,Y_1}(1,1) = 1/4 \end{array} \\[12pt] \hline & \begin{array}{ll} f_{X_2,Y_2}(0,0)= 1/2, & f_{X_2,Y_2}(0,1) = 0 \\ f_{X_2,Y_2}(1,0) = 0, & f_{X_2,Y_2}(1,1) = 1/2 \end{array} \end{align}
It follows that $f_{X_1} = f_{X_2}$ and $f_{Y_1} = f_{Y_2},$ but $X_1$ and $Y_1$ are independent but $X_2$ and $Y_2$ are not.