Proof that two utility functions $u(x_1,x_2) text{ and } v(x_1,x_2)$ with the same MRS represent the same preference

I think that you need to also assume that the utilities are non-decreasing in the goods. That is, if $x'_1>x_1$, then $u(x'_1,x_2)>u(x_1,x_2)$.

The utility functions having the "same preferences" means

$forall (x_1,x_2), (x'_1,x'_2): u(x_1,x_2) > u (x'_1,x'_2) text{ iff } v(x_1,x_2) > v(x'_1,x'_2) $

So let's assume $u(x_1,x_2) > u (x'_1,x'_2) $. Take a curve going from $(x_1,x_2)$ to a point $(x'_1, x''_2)$ by changing $x_1$ to $x'_1$ and having the second coordinate change according the MRS so that this curve has constant value of $u$ [1]. Then since $u$ is constant along this curve, $u(x'_1,x''_2) = u(x_1,x_2)$. Since $u(x_1,x_2) > u(x'_1,x'_2)$, $u(x'_1,x''_2) > u(x'_1,x'_2)$, and from the non-decreasing utility property, we have $x''_2 >x'_2$.

Since this curve was defined by the MRS of $u$, and the MRS of $u$ is equal to that of $v$, we also have that $v(x'_1,x''_2) = v(x_1,x_2)$, and so $v(x_1,x_2) > v(x'_1,x'_2)$

[1] This can be done, for example, by taking $C = (x_1+ t, x_2+int_0^tMRS_{x_1x_2}dt)$. Then taking $t=x'_1-x_1$, we get the point $left(x'_1, int_0^{x'_1-x_1}MRS_{x_1x_2}dt right)$