Given a random vector $(X_1,X_2)$. If $aX_1 + bX_2$ is Gaussian for all pairs $a,b$, then $(X_1,X_2)$ is jointly normal. More generally, is the following statement true? If $aX_1 + bX_2$ has the same distribution as $aY_1 + bY_2$, for all $a,b$, then $(X_1,X_2)$ has the same distribution as $(Y_1,Y_2)$. I know this is true if $(Y_1,Y_2)$ is the pullback under some diffeomorphism of a jointly normal vector. But is it true in general?

Also if I have total variation bound on the difference between $aX_1 + bX_2$ and $aY_1 + bY_2$ for all $a,b$, say above by $\epsilon$, does that give any total variation bound between the joint distributions?